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% for theta bits
\newcommand{\R}{\mathbb{R}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\thetachar}[2]{[#1,#2]}
\newcommand{\tfc}[2]{\tfwo\thetachar{#1}{#2}}
\newcommand{\tfcv}[4]{\tfc{#1}{#2}(#3;#4)}

\newcommand{\E}{\mathcal{E}}
\newcommand{\C}{\mathcal{C}}
\newcommand{\J}{\mathcal{J}}
\newcommand{\CK}{\mathcal{C}(K)}
\newcommand{\JK}{\mathcal{J}(K)}
\newcommand{\OO}{\mathcal{O}}
\newcommand{\PP}{{\bf P}}
%\newcommand{\wt}{\hbox{wt}}
\newcommand{\maple}[1]{\texttt{#1}}
\newcommand{\unix}[1]{\texttt{#1}}
\newcommand{\pair}[2]{(#1,#2)}
\newcommand{\ipair}[3][i]{\pair{#2_#1}{#3_#1}}
\newcommand{\embed}[1]{L(#1\Theta)}
\newcommand{\sw}[1]{\operatorname{sw}_{#1}}
\newcommand{\wred}{\operatorname{wred}\,}
\newcommand{\Symm}{S}
\newcommand{\wt}{{o_n}}
\newcommand{\ka}{\kappa}
\newcommand{\orbit}{\operatorname{orb}\:}
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\newcommand{\jac}[1]{J(\mathcal{#1})}
\newcommand{\curve}[1]{\mathcal{#1}}
\newcommand{\symmgc}[2]{S^{#1}(\curve{#2})}
\newcommand{\nThetag}[1]{\mathcal{L}(#1\Theta_g)}
\def\bbbq{{\ensuremath\mathbb Q}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \input titlepage.tex
\begin{titlepage}

\title{{\bf Hyperelliptic Curves}\\ \vspace{2cm}
Thesis submitted in accordance\\
with the requirements of the\\
University of Liverpool\\
for the degree of\\
Doctor in Philosophy}

\author{Andrew Gerard Joseph Stubbs}

\date{September 2000\footnote{{\scriptsize\ttfamily $\relax$Header: /lfs/agstubbs/cvsroot/thesis.tex,v 1.43 2001/03/20 15:32:15 agstubbs Exp $\relax$}}}

\maketitle

\end{titlepage}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \input abstract.tex
\begin{abstract}

This dissertation investigates hyperelliptic curves,
with an emphasis on curves of genus~3. The main
goal is to describe an embedding
of the Kummer Variety
in terms of the points in the support of
the divisor representing the corresponding member
of the Jacobian. Also of interest is the set
of defining equations of this variety.
The algebra grows very fast with the genus,
and a required component of this work is the
development of techniques to simplify the algebra,
and a substantial amount of programming in
algebra software packages, such as Maple.
The algorithms here will in principle apply to hyperelliptic
curves of any genus, but the size of the algebra is
such that we shall only implement them in genus~3.

\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \input ack.tex
\def\abstractname{Acknowledgements}

\begin{abstract}

I would like to thank my supervisor Victor Flynn for proposing the work in

this thesis and for his encouragement and supervision.

I would also like to thank Everett Howe and Michael Stoll for helpful

discussions by e-mail concerning some of the ideas in Section~\ref{sec:geomback}.

Thanks also to Paul Holtom, Helen Hulme, Clare Baines, Neil Kirk,

Emma Hacking, Richard Hadji, Martyn Amos, John Humphreys,

Peter Giblin, Fredrik Hagen, Jenny \"Olund, Bernard Kelly,

Elisabeth Haugvaldstad, Ayesha, Keith and Jane Garrett,

Andrew Douglas, Martin Grove, David Ashby, 

and all at Netcentric Systems PLC.

I acknowledge the Engineering and Physical Science Research Council

(EPSRC) for financial support.

\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \input dedication.tex
\begin{titlepage}

\begin{flushright}

\Large\it To my parents.

\end{flushright}

\end{titlepage}

\tableofcontents

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \input intro.tex

\chapter{Introduction}\label{ch:intro}

We shall be considering a hyperelliptic curve
of genus~$g$ defined over a field~$K$ of characteristic zero.
\begin{equation}\label{eq:hyper}
\C : y^2 = F(x) = f_n x^n + \ldots + f_0,
\end{equation}
where $n = 2g + 1$ or $2g + 2$, and $F(x)$ has
nonzero discriminant. We shall assume throughout that
either the genus is even or~$n$ is odd (or both).
Then there is an identification (up to a certain non-uniqueness)
between points
on the Jacobian and sets of~$g$ points on the curve.
More precisely, each divisor class modulo linear equivalence
contains a divisor of the form
\begin{equation}\label{eq:divisor1}
D = P_1 + \ldots + P_g - g\infty,
\end{equation}
when~$n$ is odd, or
\begin{equation}\label{eq:divisor2} 
D = P_1 + \ldots + P_g - (g/2)(\infty^+ + \infty^-),
\end{equation}
when the genus is even.
We wish to describe an embedding for the
Kummer Variety and Jacobian Variety of
a hyperelliptic curve of genus~3 into projective space,
in terms of the $x$ and $y$ coordinates of the
points $P_1, \ldots , P_g$. This is
theoretically possible, as proved in a theorem of
Lefschetz (\cite{lang:abel}, p.105), however
explicitly it is not a trivial task.

The best place to start is to examine methods used in lower genus.
To date, there are only two methods which have been employed to find
the analogous embedding for a hyperelliptic curve of genus~2. David Grant
uses $L(3\Theta)$ to obtain an embedding into $\PP^8$, but this is only useful for
quite a restricted type of genus~2 curve: $y^2=F(x)$ where $F$ is some
quintic polynomial in $x$ over the ground field. Cassels and
Flynn~\cite{prolegom} use
$L(4\Theta)$ to obtain an embedding into $\PP^{15}$ for any hyperelliptic
curve of genus~2.

Apart from the obvious restriction on the types of curve that Grant's
embedding \cite{grant:formal}
applies to, his method also suffers from depending upon there
being `enough' second derivatives of the Weierstrass $\wp$ function. This
works for genus~2, but in genus~3 we are left short. Flynn's method does
not suffer from this problem, but relies heavily on spotting relations
between symmetric polynomials in a style which is impractical
when more than~2 coordinates are involved.

A further problem that Grant's method suffers from is that he has used
the book by Baker~\cite{baker:mult} to obtain explicit expressions for
the required functions, but equivalent results for genus~3 are not so
readily available.

So far, my solution to the problem has been to simplify and generalize
the genus~2 method of Cassels and
Flynn \cite{prolegom}
as far as possible, but (when the genus~$g$ is even)
only in the slightly simpler context
of $y^2=F(x)$ where $F$ is a polynomial of degree $2g+1$ in $x$ over
the ground field.
There are a few basic expressions,
readily obtainable for any genus, from which we expect that the basis
can be obtained.

The method, using $L(n\Theta)$ to obtain an embedding of a 
hyperelliptic genus~$g$ curve
into $\PP^{n^g-1}$, is constructive
insofar as that if a symmetric polynomial in $g$
variables has the correct weight (to be defined
later), $w\leq n$, then it is either already
obtainable from previously determined basis elements, or else is
a newly discovered member. If a rational
expression $\frac{a}{b}$ is too heavy then it may be possible to reduce
the weight by considering $a$ `mod' $b$. In univariate expressions this is
obviously an application of the Euclidean Algorithm, but in
multivariate expressions, there is no unique way of tackling the problem.
The approach will be first to define a set of rational
functions in $x_1,\ldots x_g, y_1,\ldots , y_g$, consisting
of the $a_i$ , which span the regular symmetric functions
in $x_1,\ldots x_g$, and the $b_i$ are the coefficients of
the polynomial passing through the $(x_i, y_i)$.
One then searches for polynomials in the $a_i, b_i$
which give a basis for a given $L(n\Theta)$.
We shall use this first to show how the genus~2 objects
in~\cite{prolegom} can be more simply derived,
and then proceed to genus~3.

I started by examining symmetric
polynomials, developing a couple of algorithms;
one to convert a given symmetric polynomial in 3 variables
into an equivalent expression in terms of `elementary' symmetric
polynomials, and another more complex algorithm to perform the same
task for 3 pairs of variables. At this point it appears that this is
in fact only necessary to provide `simplified' expressions for
human-readable output, or possibly to reduce the workload (and hence
CPU time) for other algorithms used in the process.

I have also implemented a multivariate division algorithm, as described
in Cox, Little, O'Shea. This is the canonical generalization of the
Euclidean Algorithm, and will be used especially in reducing
weights as described above. I have made my approach as structured as
possible in terms of
pre-existing computational symbolic algebra and have avoided reinventing
the wheel wherever possible, which has involved some research into this
not-directly-related field.

I have attempted to use quite a few programming languages to implement
these algorithms; C \& PARI~\cite{pari}, Java, Maple V R3,
KASH~\cite{kant} and~\cite{bruin:ellshar}, amongst a host of other
less successful attempts. At the moment I have settled on Maple, but I do
not particularly like the language and am continually vexed by the
policy of Waterloo Software in releasing backwards-incompatible versions
of Maple V. The computing situation, though greatly improved
over the past few years, is still far from ideal.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \input geometry.tex

\chapter{Geometry}\label{ch:geometry}

We shall describe some general ideas concerning the theta
divisor, and the embeddings of $L(n\Theta)$ which
can be used to give the Kummer and Jacobian as
projective varieties. 

\section{Geometric Background}\label{sec:geomback}

It is instructive first to consider what happens in genus~1
and genus~2. First consider the elliptic curve

\begin{equation}\label{eq:ell}
\E : y^2 = x^3 + g_2 x^2 + g_1 x + g_0,
\end{equation}

\noindent defined over a field~$K$ of characteristic zero,
where the cubic has nonzero discriminant.
We shall take $\infty$ to be the identity. There is
the standard correspondence between points on~$\E$
and members of the Jacobian, where $P \in \E$ corresponds
to the divisor class containing $P - \infty$.
Define the standard weights function $\wt (f)$ by considering the order of
the pole of $f$ at $\infty$ on the curve; on $\E$, define

\begin{equation}\label{eq:ellwt}
\wt (x) = 2,\ \wt (y) = 3.
\end{equation}

Now, consider the bases for functions of weight at
most~2, at most~3 and at most~4:

\begin{equation}\label{eq:ellkumrhoz}
\begin{array}{rcl}
&\ka_0 = 1,\ \ka_1 = x,\\
&\rho_0 = 1,\ \rho_1 = x,\ \rho_2 = y,\\
&z_0 = 1,\ z_1 = x,\ z_2 = y,\ z_3 = x^2.\\
\end{array}
\end{equation}

We can use either the second or third bases to
give the coordinates of a projective embedding
of the Jacobian variety. In the case of $\rho_0, \rho_1, \rho_2$,
we have as the defining equation merely the homogenisation
of~(\ref{eq:ell}).

\begin{equation}\label{eq:ellrho}
\E : \rho_0 \rho_2^2 = 
\rho_1^3 + g_2 \rho_0 \rho_1^2 + g_1 \rho_0^2 \rho_1 + g_0 \rho_0^3.
\end{equation}

In the case of $z_0, z_1, z_2, z_3$,
we have a pair of defining equations

\begin{equation}\label{eq:ellz}
\begin{array}{rcl}
&z_2^2 = z_1 z_3 + g_2 z_0 z_3 + g_1 z_0 z_1 + g_0 z_0^2,\\
&z_1^2 = z_0 z_3,\\
\end{array}
\end{equation}

\noindent which give the Jacobian as the intersection of
quadratic forms, as a special case of~\cite{mumford:abelian}.
Note that here, with genus~$g=1$, there are  
$2^g, 3^g, 4^g$ members of the bases in~(\ref{eq:ellkumrhoz}),
and that the defining equations for the~$\rho_i$ are
cubic, and the defining equations for the~$z_i$ are quadratic.
Also, note that, of the $z_i$, $1 = (4^g - 2^g)/2$ is {\em odd} (negated
by $P \mapsto -P$) and $3 = (4^g + 2^g)/2$ are
{\em even} (invariant under $P \mapsto -P$).

In genus~2, let

\begin{equation}\label{eq:genus2}
\C : y^2 = f_6 x^6 + f_5 x^5 + \ldots + f_0,
\end{equation}

\noindent defined over a field~$K$ of characteristic zero,
where the sextic has nonzero discriminant.
When~$f_6 \not= 0$,
we use~$\infty^+, \infty^-$ to denote
the points on the non-singular curve, the normalisation of $\C$,
that lie over the
singular point at infinity on~$\C$
(for any curve~(\ref{eq:genus2}) with~$f_6\not=0$
both~$\infty^+$ and~$\infty^-$ are in~$\CK$ when~$f_6 \in (K^*)^2$).
When~$f_6 = 0$ (and so~$f_5 \not= 0$),
we let~$\infty$ denote the point at infinity,
which is always in~$\CK$.
Following Chapter~1 of~\cite{prolegom},
any member of~$\JK$, the $K$-rational points on the~Jacobian,
may be represented by a divisor of the form
$P_1 + P_2 - \infty^+ - \infty^-$, where
$P_1,P_2$ are points on~$\C$ and either
$P_1,P_2$ are both $K$-rational or
$P_1,P_2$ are quadratic over~$K$ and conjugate.
We shall abbreviate such a divisor by: $\{ P_1 , P_2 \}$.
This representation gives a 1-1 correspondence with
members of~$\JK$, except that everything of the form $\{ (X,Y),(X,-Y)\}$
must be identified into a single equivalence class
$\OO$, which serves as the group identity in $\JK$.
Note that
$-\{  (x_1,y_1), (x_2,y_2) \} = \{  (x_1,-y_1), (x_2,-y_2) \}$;
furthermore 
$\{ P_1, P_2 \} + \{ Q_1, Q_2 \} + \{ R_1 , R_2 \} = \OO$
if and only if there exists $\Upsilon(X)$ of degree~$\le 3$ such that
$Y = \Upsilon(X)$ meets~$\C$ at $P_1,P_2,Q_1,Q_2,R_1,R_2$. 
These two rules are sufficient for
computing the group law on~$\JK$. Clearly, an element of order~2
in~$\JK$ is given by $\{ (X_1, 0), (X_2, 0) \}$, where
$X_1, X_2$ are the roots of quadratic $Q(X)$
defined over~$K$, satisfying $Q(X) | F(X)$.

We define

\begin{equation}\label{eq:gen2wt}
\wt (x) = 2,\ \wt (y) = \hbox{ 5 (if $f_6 = 0$) or 6 (if $f_6 \not= 0$)}.
\end{equation}

Following a similar route to that exposed in
Cassels and Flynn~\cite{prolegom}, we first look at
functions in $x_1,y_1,x_2,y_2$
regular away from $\OO$ with order of pole at infinity of at most~$2$;
that is to say, a set of coordinates for
the {\em Kummer surface}.
By this we mean that, within each monomial, we consider
the weight in $(x_1, y_1)$ and $(x_2,y_2)$ separately,
and take the maximum. For example, the weight (or order
at infinity) of $x_1^3 x_2$ is~$6$ (since the weight
of $x_1^3$ is~$6$ and that of $x_2$ is~$2$).
For rational functions, the weight is just the
weight of the numerator minus that of the denominator. 
We can also think of this as a basis for $L(\Theta^+ + \Theta^-)$,
where $\Theta^+$ and $\Theta^-$ are the images of the
curve in the Jacobian given by the images of
$P \mapsto P - \infty^+$ and $P \mapsto P - \infty^-$, respectively,
These are given in Chapter~2 of~\cite{prolegom} by

\begin{equation}\label{eq:gen2kum}
\ka_0=1,\ \ka_1=(x_1+x_2),\ 
\ka_2=x_1 x_2,\ \ka_3={{F_0(x_1,x_2)-2y_1 y_2}\over{(x_1-x_2)^2}},
\end{equation}

\noindent where 

\begin{equation}\label{eq:f0}
\begin{array}{rcl}
F_0(x_1,x_2)=\ &2f_0+f_1(x_1+x_2)+2f_2(x_1 x_2)+f_3(x_1+x_2)(x_1 x_2)\\
& +2f_4(x_1 x_2)^2
+f_5(x_1+x_2)(x_1 x_2)^2+2f_6(x_1 x_2)^3.\\
\end{array}
\end{equation}

Note that there are $4 = 2^g$ members of this basis
$\ka_0, \ldots , \ka_3$ of $L(\Theta^+ + \Theta^-)$
when $f_6\not=0$, and the same definitions of $\ka_0, \ldots , \ka_3$ 
give a basis for $L(2\Theta)$ when $f_6 = 0$.

When~$f_6 = 0$, Grant~\cite{grant:formal} has described a basis
$\rho_0, \ldots , \rho_8$ for $L(3\Theta)$, which
we do not reproduce here. This gives an embedding
of the Jacobian variety in $\PP^8$. Again note that
there are $9 = 3^g$ members of this basis for $L(3\Theta)$.

When~$f_6 \not= 0$, $L(3\Theta)$ is not defined over
the ground field, and we must use $L(2(\Theta^+ + \Theta^-))$,
which is geometrically the same as $L(4\Theta)$
when $f_6 = 0$. We reproduce here the basis $z_0,\ldots , z_{15}$ in
Chapter~2 of~\cite{prolegom}.

\begin{equation}\label{eq:gen2z}
\begin{array}{rcl}
&z_{15} = (x_1-x_2)^2,\ 
z_{14} = 1,\ z_{13} = x_1 + x_2,\ z_{12}=x_1 x_2 ,\\
&z_{11} = x_1 x_2 (x_1 + x_2 ),\
z_{10} = (x_1 x_2 )^2,\
z_9 =(y_1 -y_2 )/(x_1 -x_2 ),\\
&z_8 =(x_2 y_1 -x_1 y_2 )/(x_1 -x_2 ),\
z_7 =(x_2 ^2 y_1 - x_1 ^2 y_2 )/(x_1 -x_2 ),\\
&z_6 =(x_2 ^3 y_1 -x_1 ^3 y_2 )
/(x_1 -x_2 ),\
z_5 = (F_0 (x_1 ,x_2 )-2y_1 y_2 )/(x_1 -x_2 )^2,\\
&z_4 = (F_1 (x_1 ,x_2 )-(x_1 +x_2 )y_1 y_2 )/(x_1 -x_2 )^2,\
z_3 = (x_1 x_2 )z_5,\\
&z_2 = (G_0 (x_1 ,x_2 )y_1 - G_0 (x_2 ,x_1 )y_2 )/(x_1 - x_2 )^3,\\
&z_1 = (G_1 (x_1 ,x_2 )y_1 - G_1 (x_2 ,x_1 )y_2 )/(x_1 - x_2 )^3,\
z_0 = z_5^2.\\
\end{array}
\end{equation}

\noindent where

\begin{equation}\label{eq:f0f1}
\begin{array}{rcl}
F_0 (x_1 ,x_2 )= &2 f_0 + f_1 (x_1 +x_2 ) + 2 f_2 (x_1 x_2 )
+ f_3 (x_1 x_2) (x_1 +x_2 ) \\
&+ 2 f_4 (x_1 x_2 )^2 + f_5 (x_1 x_2 )^2 (x_1 + x_2 )
+2 f_6 (x_1 x_2 )^3,\\
F_1 (x_1 ,x_2 )= &f_0 (x_1 +x_2 ) + 2 f_1 (x_1 x_2 )+f_2 (x_1 x_2 )(x_1 +x_2)
 + 2 f_3 (x_1 x_2 )^2 \\
&+ f_4 (x_1 x_2 )^2 (x_1 +x_2 ) + 2 f_5 (x_1 x_2)^3
 + f_6 (x_1 x_2 )^3 (x_1 +x_2 ),\\
\end{array}
\end{equation}
\begin{equation}\label{eq:g0g1}
\begin{array}{rcl}
G_0 (x_1 ,x_2 )= &4 f_0 + f_1 (x_1+3 x_2)+f_2(2x_1 x_2+2 x_2 ^2 )
   + f_3(3x_1 x_2 ^2 + x_2 ^3 ) \\
&+4 f_4(x_1 x_2 ^3)+f_5(x_1 ^2 x_2 ^3 + 3 x_1 x_2 ^4)
  + f_6(2 x_1 ^2 x_2 ^4 + 2 x_1 x_2 ^5 ),\\
G_1 (x_1 ,x_2 )= &f_0(2x_1+2x_2)+f_1(3x_1 x_2 + x_2 ^2) + 4f_2(x_1 x_2 ^2)
  + f_3(x_1 ^2 x_2 ^2 +3x_1 x_2^3)\\
&+f_4(2x_1^2 x_2^3 +2x_1 x_2^4) +f_5(3x_1^2 x_2^4 +x_1 x_2^5)
+4f_6(x_1^2 x_2^5).\\
\end{array} 
\end{equation}

There are $16 = 4^g$ members of this basis, which also
gives an embedding of the Jacobian variety.
Note that $6 = (4^g - 2^g)/2$ are {\em odd} (negated
by $P \mapsto -P$) and $10 = (4^g + 2^g)/2$ are
{\em even} (invariant under $P \mapsto -P$).
The canonical
divisor class $\OO$ is mapped to $(1,0,\ldots ,0)$,
and the Mordell-Weil group $\hbox{Pic}^0_K(\C)$
is mapped into ${\bf P}^{15}(K)$.

The defining equation for the Kummer surface
is given by the quartic equation in $\PP^3$,

\begin{equation}
\label{eq:kummdefn}
 R(\ka_0,\ka_1,\ka_2) \ka_3^2 + S(\ka_0,\ka_1,\ka_2) \ka_3 + T(\ka_0,\ka_1,\ka_2) = 0,
\end {equation}
where $R$, $S$, $T$ are given by:
\begin {eqnarray*}
R(\ka_0,\ka_1,\ka_2) &=& \ka_1^2 - 4 \ka_0 \ka_2, \\
S(\ka_0,\ka_1,\ka_2) &=&
-2 \left( 2 f_0 \ka_0^3 + f_1 \ka_0^2 \ka_1 + 2 f_2 \ka_0^2 \ka_2 + f_3 \ka_0 \ka_1 \ka_2
+ 2 f_4 \ka_0 \ka_2^2 \right. \\
  &~& \left. + f_5 \ka_1 \ka_2^2 + 2 f_6 \ka_2^3 \right), \\
T(\ka_0,\ka_1,\ka_2) &=&
- 4 f_0 f_2 \ka_0^4 + f_1^2 \ka_0^4 - 4 f_0 f_3 \ka_0^3 \ka_1 - 2 f_1 f_3 \ka_0^3 \ka_2 
- 4 f_0 f_4 \ka_0^2 \ka_1^2 \\
&~& + 4 f_0 f_5 \ka_0^2 \ka_1 \ka_2 - 4 f_1 f_4 \ka_0^2 \ka_1 \ka_2 
- 4 f_0 f_6 \ka_0^2 \ka_2^2
+ 2 f_1 f_5 \ka_0^2 \ka_2^2 \\
&~& - 4 f_2 f_4 \ka_0^2 \ka_2^2 + f_3^2 \ka_0^2 \ka_2^2 - 4 f_0 f_5 \ka_0 \ka_1^3 
+ 8 f_0 f_6 \ka_0 \ka_1^2 \ka_2 - 4 f_0 f_6 \ka_1^4  \\
&~& - 4 f_1 f_5 \ka_0 \ka_1^2 \ka_2 + 4 f_1 f_6 \ka_0 \ka_1 \ka_2^2
- 4 f_2 f_5 \ka_0 \ka_1 \ka_2^2 - 2 f_3 f_5 \ka_0 \ka_2^3 \\
&~& - 4 f_1 f_6 \ka_1^3 \ka_2 - 4 f_2 f_6 \ka_1^2 \ka_2^2
-4 f_3 f_6 \ka_1 \ka_2^3 - 4 f_4 f_6 \ka_2^4 + f_5^2 \ka_2^4.
\end {eqnarray*}

The defining equations for Grant's $L(3\Theta)$ embedding are
given by equations that are at most cubic, and the
defining equations for the $L(4\Theta)$ embedding
are give by quadratic equations. 

We now generalise the above patterns
and definitions, and give some standard 
results about dimensions and embeddings.

\begin{define}\label{def:theta}
For $\C$ as in~(\ref{eq:hyper}) with~$n$ odd, define $\Theta$ to be
the set of all divisors of the forms: $(x_1,y_1)-\infty,$
$(x_1,y_1)+(x_2,y_2)-2\infty,$ $\ldots$,
$(x_1,y_1)+..+(x_{g-1},y_{g-1}) - (g-1)\infty$, as described in
\cite{LB},~11.2.
\end{define}

When the genus is~2, $L(2\Theta)$ gives an embedding of the
{\em Kummer surface}. It is natural to generalise this
as follows.

\begin{define}\label{def:kummer}
For any hyperelliptic curve $\C$, we shall
refer to the locus of a basis of $L(2\Theta)$
as a {\em Kummer variety}.
\end{define}

By \cite{LB}, 7.3, 7.4, 11.1 and
\cite{mumford:abelian}, p.163, we have the following.

\begin{lemma}\label{lem:ample}
Let $\Theta$ be as in Definition~(\ref{def:theta}).
Then $\Theta$ is ample, and $L(n\Theta)$ for $n\ge 3$ will be very ample, and
so induces a projective embedding of the Jacobian.
The defining equations for the $L(3\Theta)$ embedding of the Jacobian
variety are all at most cubic, and the
defining equations for the $L(4\Theta)$ embedding of the Jacobian
variety are all quadratic.
\end{lemma}

By analogy with genus~2, where the defining equation is a quartic
(perhaps provable by generalising the argument in~10.3,
although we have not been able to do so), we make the following
conjecture.

\begin{conj}\label{conj:kum}
Let $\Theta$ be as in Definition~(\ref{def:theta}).
The defining equations for the
$L(2\Theta)$ embedding of the Kummer variety
are given by equations of degree at most four.
\end{conj}

Note that this is compatible with the fact that the defining
equations for the $L(4\Theta)$ embedding are all quadratic,
since any such quadratic equation in the even functions
of $L(4\Theta)$ induces a quartic
equation in $L(2\Theta)$. 

\begin{lemma}\label{lem:dimLnT}
(Flynn~\cite{flynn:g2arbitrary}, Grant~\cite{grant:formal}) The
dimension of $L(n\Theta)$ is $n^g$ in genus~$g$.
\end{lemma}

\begin{lemma}\label{lem:oddeven}
(\cite{LB}, pp. 57,58,66,93)
There is a basis of $L(4\Theta)$ consisting of
${1\over 2}(4^g + 2^g)$ even functions
and ${1\over 2}(4^g - 2^g)$ odd functions.
\end{lemma}

A natural problem now is to try to find objects analogous to
the $\ka_i$, $\rho_i$ and $z_i$ above, in the
case of genus~3. One can see from the above how quickly
the quantity of algebra explodes in going from genus~1 to~2,
and there is a similar further explosion in the size
of the expression in going from genus~2 to genus~3.
Therefore there will be severe
computational constraints on what
is possible.

\section{Theta Functions}\label{sec:theta}

Our techniques will not rely on the theory of theta
functions. Nevertheless, for the sake of context,
we shall give a brief overview,
broadly following \cite{baker:mult} but with
the notation slightly modernised. 
We shall assume, as is common
in analytic treatments, that the curve (\ref{eq:hyper})
is in `canonical' form, with $f_{2g+2} = 0$
and $f_{2g+1} = 4$.

%% FIXME
Remember that, for a given hyperelliptic curve, $\mathcal{C}$, of
genus $g$, there is a lattice $\Omega_\mathcal{C}$ such that
$\mathbb{C}^g/\Omega_\mathcal{C}$ is homeomorphic, by Abel's Theorem,
to the Jacobian of $\mathcal{C}$.

\begin{define}
 A theta function on this lattice is a holomorphic
 function which is pseudo-periodic with respect to the lattice $\Omega$,
 often defined as
 \begin{equation}
  \vartheta(z,\Omega)=\sum_{n\in\Z^g}\exp (\pi i n^{t}\Omega n + 2\pi i n^t z)
 \end{equation}
\end{define}

A slight generalization of $\vartheta$ are the theta functions with
rational characteristics $\vartheta\thetachar{a}{b}(z,\Omega)$ with
$a,b\in\Q^g$.

Riemann has shown that there are $a,b,\in\Z$ s.t.  $\vartheta[a,b](z)$
has a zero of order one precisely along the pullback of $\Theta$ to
$\C^g$. This is the specific theta function, $\vartheta$ which we are
particularly interested in, as the second derivatives of $\ln
\vartheta$ give rise to the function known as the Weierstrass
$\wp$-function. We denote
$\zeta_i={\partial \ln \vartheta \over \partial u_i}$, where $u_i$ is
a coordinate on $\mathbb{C}^g/\Omega_\mathcal{C}$.

We give the following inductive
definition.

\begin{define}\label{def:pij}
\begin{equation}\label{eq:pij}
\begin{array}{rcl}
\wp_{ij}(u) &= -{\partial \zeta_i \over \partial u_j} (u)\\
\wp_{i\ldots jk} &= -{\partial \over \partial u_k} \wp_{i\ldots j}(u)\\
\end{array}
\end{equation}
\end{define}

For notational convenience we also make the following
definition.

\begin{define}\label{def:lst}
\begin{equation}\label{eq:lst}
\ell_i (s,t) = {1\over 4t}\sum_{k=i}^{2g+1-i} (k+1-i) f_{k+1+i} s^k,
1 \leq i \leq g.
\end{equation}
\end{define}

\begin{lemma}\label{lem:canonbasis}
The canonical basis for the differentials of the second kind is
given by $\ell_i (s,t)\hbox{d}s$, where $t^2 = F(s)$. 
\end{lemma}

\begin{proof}
See \cite{baker:abel} \S 138, especially Ex.\ (i).
\end{proof}

\begin{define}\label{def:intsec}
For $1 \leq i \leq g$ and $t^2 = F(s)$ we define the integrals
of the second kind $L_i$:
\begin{equation}\label{eq:intsec}
L_i^{P,Q} = \int_Q^P \ell_i (s,t)\hbox{d}s.
\end{equation}
\end{define}

\begin{lemma}\label{lem:convexp}
Using the convention that ${\partial x_k \over \partial u_i} = 0$
for $i > g$ we find the following convenient expression
\begin{equation}\label{eq:convexp}
\zeta_i(u) = - \sum_{k=1}^g L_i^{P_k,Q_k}
            + {1\over 2} \sum_{k=1}^g {\partial x_k \over \partial u_{i+1}}
\end{equation}
\end{lemma}

\begin{proof}
See \cite{baker:abel} \S 215, especially Ex.\ (vi).
\end{proof}

\begin{lemma}\label{lem:ratfn}
We can express the hyperelliptic $\wp$ functions as rational
functions of the variables $x_1,\ldots ,x_g,y_1,\ldots ,y_g$
on $S^g\C$.
\begin{equation}\label{eq:ratfn}
\wp_{ij}(u) = \sum_{k=1}^g \ell_i(x_k,y_k){\partial x_k \over \partial u_j}
   - {1\over 2}\sum_{k=1}^g {\partial^2 x_k \over \partial u_j \partial u_{i+1}}.
\end{equation}
\end{lemma}

\begin{proof}
Differentiate (\ref{eq:convexp}) with respect to $u_j$.
\end{proof}

In order to calculate $\partial x_k \over \partial u_i$ we need to work
with the system of differential equations

\begin{equation}\label{eq:sysdiff}
\hbox{d}u_i = \sum_{k=1}^g x_k^{i-1} {\hbox{d}x_k \over y_k},
1\leq i \leq g,
\end{equation}

\noindent Define~$V$ to be the Vandermonde matrix
for $x_1,\ldots , x_g$ to obtain

\begin{equation}\label{eq:vand}
(\hbox{d}u_1,\ldots , \hbox{d}u_g)
= \Bigl( {\hbox{d}x_1 \over y_1}, \ldots , {\hbox{d}x_g \over y_g}\Bigr) V.
\end{equation}

\noindent $V$ is nonsingular, and so we can express
$\frac{\text{d}x_i}{y_i}$ in terms of the $\hbox{d}u_j$.
Hence we have the following lemma.

\begin{lemma}\label{lem:vandeqn}
Define~$V$ to be the Vandermonde matrix for
$x_1,\ldots ,x_g$. Then, denoting the ith row and
jth column in~$V^{-1}$ by $V_{i,j}^{-1}$ we have
\begin{equation}\label{eq:vandeqn}
{\partial x_i \over \partial u_j} = y_i V_{j,i}^{-1}.
\end{equation}
\end{lemma}

In \cite{grant:formal} second derivatives of logs
of theta functions are used in genus~2 to
get bases for such objects as $L(2\Theta)$.
In genus~3, it must be possible in principle
to use theta functions, at least in the $y^2 = \hbox{ odd degree}$
case, but it seems to be much less straightforward
than in genus~2 to find bases expressed in terms
of the~$x_i$ and~$y_i$. For example, in genus~3
there is no way to determine $L(2\Theta)$ which has dimension~8 purely
from second derivatives of the log of a theta function, of which there
are~6. One needs some fairly exotic polynomial in these 6 functions to
fill out $L(2\Theta)$. Our approach will be entirely independent
of theta functions, and will rely essentially only
on linear algebra.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \input kumjac.tex

\chapter{Kummer and Jacobian Varieties for Curves of Genus 3}
\label{ch:kumjac}
% L(2*Theta), L(3*Theta).
%Big section on weights (various kinds). Coeffs of (y1-y2), etc.
%Symmetric functions. Denominator delta, etc.

We shall consider the problem of trying to find a basis
for $L(n\Theta)$ in terms of the $x_i$, $y_i$ that
occur in the support of $(x_1,y_1) + \ldots + (x_g,y_g) - g\cdot \infty$. 
More detailed descriptions of some of the algorithms
will be in the next chapter.
We shall first require some preliminaries on symmetric polynomials.
We then proceed to define some weights which are
required to simplify computations. We show how to
obtain an elementary rederivation of the basis elements
in genus~2, and then give our main result: a
basis for $L(2\Theta)$ in genus~3.

\section{Outline of General Strategy}\label{sec:strategy}

For a set of~$g$ points $(x_1,y_1),\ldots ,(x_g,y_g)$
on a hyperelliptic curve~(\ref{eq:hyper}) of genus~$g$, define
$a_i$ and $b_i$ as follows.

\begin{define}\label{def:ab}
Let $a_0,\ldots ,a_g$ be the unique polynomials
in $x_1,\ldots ,x_g$ satisfying
\begin{equation}\label{eq:adefn}
(x + x_1)(x + x_2)\ldots (x + x_g) = 
   a_0 x^g + a_1 x^{g-1} + \ldots + a_g.
\end{equation}
Let $b_0,\ldots ,b_{g-1}$ be the unique rational
functions in $x_1,\ldots ,x_g,y_1,\ldots ,y_g$ satisfying
\begin{equation}\label{eq:bdefn}
y_i = b_0 x_i^{g-1} + b_1 x_i^{g-2} + \ldots + b_{g-1},
\hbox{ for all $i$ satisfying }1\leq i \leq g.
\end{equation}
\end{define} 

The polynomials $a_0,\ldots, a_g$ are often referred to
as the {\em elementary symmetric polynomials}
in $x_1, \ldots x_g$. Any symmetric polynomial in
$x_1, \ldots x_g$ is a polynomial in the elementary symmetric
polynomials. The rational functions $b_0,\ldots b_g$
have denominator

\begin{equation}\label{eq:delta}
\delta = \hbox{det} \bigl( x_j^{i-1} \bigr).
\end{equation}

An affine piece of the Jacobian can be represented
by the $(2g-1)$-tuples $[a_0,\ldots ,a_g,b_0,\ldots ,b_{g-1}]$.
However, this way of representing members of the Jacobian
does not immediately give us all of the members of
a given $L(n\Theta)$. To take a simple example, when~$g=2$,
we know from Lemma~\ref{lem:dimLnT} that the
dimension of $L(2\Theta)$ is~4, whereas only $a_0,a_1,a_2$
(and no other $a_i$ and no $b_i$) are members of $L(2\Theta)$,
so that one basis member is missing. For higher genus
and higher values of~$n$, the situations becomes worse,
with progressively smaller and smaller proportions
of the $n^g$ basis members for $L(n\Theta)$ appearing
directly on the list $a_0,\ldots ,a_g,b_0,\ldots ,b_{g-1}$.

Our strategy is to obtain missing members of a given
$L(n\Theta)$ as polynomials in $a_0,\ldots ,a_g,b_0,\ldots ,b_{g-1}$.
We first generalise the weight of (\ref{eq:ellwt}),(\ref{eq:gen2wt}).

\begin{define}\label{def:wt}
Let $\C : y^2 = f_n x^n + \ldots + f_0$ be a hyperelliptic curve of
genus~$g$ as in (\ref{eq:hyper}).
Define $\wt (x_i) = 2$, $\wt (y_i) = n$ for any~$i$,
and $\wt (f_j) = 0$.
Define $\wt$ on monomials by
\begin{equation}\label{eq:wtmono}
\wt (x_1^{i_1} \ldots x_g^{i_g} y_1^{j_1} \ldots y_g^{j_g})
= \hbox{max}\bigl( 2 i_1 + n j_1, \ldots , 2 i_g + n j_g \bigr).
\end{equation}
Define $\wt$ of a polynomial in $x_1,\ldots ,x_g,y_1,\ldots ,y_g$
to be the maximum $\wt$ of a monomial occurring in the polynomial.
Extend $\wt$ to rational functions in $x_1,\ldots ,x_g,y_1,\ldots ,y_g$  
by defining $\wt (p/q) = \wt(p) - \wt(q)$.
\end{define}

To find the missing members of a given $L(n\Theta)$,
we seek polynomials in $a_0,\ldots ,a_g,b_0,\ldots ,b_{g-1}$
which have weights at most~$n$.
Since the expressions in the $x_i$, $y_i$ are extremely large,
we shall require ways of simplifying the computations,
which will be described and illustrated in the following sections.

\section{Symmetric polynomials in pairs of variables}\label{sec:sympoly}

The notion of symmetric and antisymmetric polynomials in pairs of
variables is crucial in much of the work of pole-reduction. Here
we examine some of the general theory which is required.

\begin{define}
 A \emph{symmetric polynomial in pairs of variables} is a polynomial in
 a given number, $n$, of pairs of variables $(x_i,y_i)$ which is independent
 under the action of the permutation group $S_n$ given by
 $\sigma:(x_i,y_i)\mapsto (x_{i\sigma},y_{i\sigma})$. We denote the space
 of such functions $S_n\mathbb{F}[X,Y]$; a subspace of
 $\mathbb{F}[X_1,\ldots,X_n,Y_1,\ldots,Y_n]$.
\end{define}

\begin{define}
 Define an index function 
 \[i_n(x,y):\{(x,y):1\leq x+y\leq n\}
  \rightarrow\!\!\!\!\!\rightarrow \{1,\ldots,\frac{1}{2}n(n+3)\}\]
 as follows:
 \[i_n(x,y)=\frac{y(2n+3)-y^2}{2}+x\]
 This is invertible in the range of definition, with inverse given by:
 \[i_n^{-1}(m)=(m-i_n(0,y),y)\]
 where $0\leq y\leq n$ is chosen so that either:
 \begin{itemize}
  \item $y=n$ and $i_n(0,y)= m$; or
  \item $i_n(0,y) \leq m < i_n(0,y+1)$, for $y<n$.
 \end{itemize}
\end{define}

\begin{define}
 Define \emph{elementary symmetric polynomials} in the $n$ pairs of variables
 $(x_i,y_i)$, as follows $(1\leq x+y\leq n)$:
 \[
  r_{i_n(x,y)} = \sum_{p\in I_y}
     \bigg((\prod_{j\in p} Y_j)(\sum_{q\in J_{p,x}}\prod_{k\in q} X_{k})\bigg)
 \]
 where $I_y=\{\{1\sigma,\ldots,y\sigma\}:\sigma\in S_n\}$,
 $I_0=\{\emptyset\}$, and
 $\forall p\in I_y, J_{p,x}=\{(\{1,\ldots,n\}-p)\hbox{ choose }x\}$.
\end{define}

\begin{conj}
 Any polynomial symmetric in the $n$ pairs of variables $(x_i,y_i)$,
 $1\leq i\leq n$, may be expressed in terms of the elementary
 symmetric polynomials $r_j$, $1\leq j\leq \frac{1}{2}n(n+3)$.
\end{conj}

\begin{lemma}
 The weight functions $x(r_j)$ and $y(r_j)$ are given as follows:
 \begin{align*}
  x(r_j) &= \pi_x i_n^{-1}(j)\\
  y(r_j) &= \pi_y i_n^{-1}(j)\hbox{.}
 \end{align*}
\end{lemma}

Then each monomial in $r_j$ will have
a well-defined weight-pair $(x,y)$ which allows us to
partition the space of symmetric polynomials in the
$(x_j,y_j)$ into homogeneous subspaces; any polynomial
expression
in the $r$ may be projected onto those spaces to
obtain $(x,y)$-homogeneous subexpressions which when added
together give the original expression.

\begin{define}
 Denote by $H_{x,y}$ the homogeneous subspace of $S_n\mathbb{F}[X,Y]$ of
 weight $(x,y)$. Also use this symbol to denote the projection map
 $H_{x,y}:S_n\mathbb{F}[X,Y]\rightarrow\!\!\!\!\!\rightarrow H_{x,y}[X,Y]$.
 Clearly,
 \[S_n\mathbb{F}[X,Y]=\bigoplus_{x,y\geq 0}H_{x,y}[X,Y]\hbox{.}\]
\end{define}

\section{Representation of Symmetric Polynomials}\label{sec:symrep}

Given a symmetric polynomial $S$ in variables $x_1,\ldots,x_g$, and
limited processing power, we soon find ourselves looking for the
minimum amount of information required such that the original
$S$ may be reconstructed as and when necessary. The initial approach
may be to try to express the polynomial $S$ in terms of the 
appropriate elementary symmetric polynomials, but this
quickly runs into trouble for precisely the reasons that we are
trying to prevent.

\begin{example}
Given $S=x_1^3(x_2+x_3)+x_2^3(x_1+x_3)+x_3^3(x_1+x_2)$, then
$S$ may be reconstructed from $x_1^3x_2$.
\end{example}

Suppose the degree of $S$ with respect to $x_1$ alone is $d$. Then
the coefficient of $x_1^d$ is symmetric in $x_2,\ldots,x_g$. This
gives rise to a recursive procedure for completely describing at least
the terms in $S$ containing $x_i^d$ for each $i$.

This recursivity pretty much completely describes the algorithm
required for `normal' symmetric polynomials, and suggests a canonical
representation for $S$:
\[
 c_1 x_1^{d_{1,1}}x_2^{d_{1,2}}\ldots x_g^{d_{1,g}}
  + \cdots
  +c_k x_1^{d_{k,1}}x_2^{d_{k,2}}\ldots x_g^{d_{k,g}}
\]
where $d_{i_1,j_1}\geq d_{i_2,j_2}$ when either $i_1<i_2$ and $j_1=1=j_2$;
or $i_1=i_2$ and $j_1<j_2$. If $C_1$ and $C_2$ are the
canonical representations of $S_1$ and $S_2$ respectively, then
$C_1+C_2$ is the canonical representation for $S_1+S_2$.

\section{Representation of Hyperelliptic Expressions}\label{sec:rephyper}

Recall that, in this context, a Hyperelliptic Expression is a
rational polynomial symmetric in pairs of variables $(x_i,y_i)$, 
$1\leq i \leq g$ such that the highest power of any individual
$y_i$ is exactly one.

This gives us a natural representation:
\[
 A=A_0 + A_{1,1}y_1 + \cdots + A_{1,g}y_g + A_{2,1} y_1y_2 + \cdots 
   + A_{2,\frac{g!}{2!(g-2)!}} y_{g-1}y_g + \cdots
\]
where $A_{i,1}$ is symmetric in $\{x_1,\ldots,x_i\}$ and
$\{x_{i+1},\ldots,x_g\}$ separately. The conditions require
that the orbit of $A_{i,1}y_1\ldots y_i$ is symmetric in the pairs
$(x_i,y_i)$ for all $1\leq i\leq g$.

Determine the representative, $R_{i,1}$, for $A_{i,1}$ regarded as a
symmetric polynomial in $\{x_1,\ldots,x_i\}$. Then, find
the representative $R_2$ for $A_{i,1}-\orbit R_{i,1}$ regarded as a
symmetric polynomial in $\{x_{i+1},\ldots,x_g\}$. Then,
the representative for the terms with $y$-weight $i$ is simply
$R_i=(R_{i,1}+R_{i,2})y_1\ldots y_i$.

Denoting the canonical representative of $A_0$ by $R_0$ gives a canonical
representation:
\[
 A = \orbit \sum_j R_j
\]

\section{Weights and Measures}\label{sec:weights}

The techniques I have developed require us to treat expressions in the
$x$'s, $y$'s and $f$'s as a graded pseudo-homogeneous ring. The
weights of principal importance are as indicated in Figure~\ref{fig:wam}.
Note that, in the figure the $x$, $y$, $w$ columns are weights and
the $o$ columns are the order of the pole at $\infty$ for that
expression. Also, $w=(g+1)y+x$.

\begin{figure}
\begin{center}
 \begin{tabular}{|r||c|c|c|c|c|}
  \hline           &  $x$  & $y$ & $w$      & $o_{2g+1}$ & $o_{2g+2}$    \\
  \hline $x_i$     &  $1$  & $0$ & $1$      &  $2$   & $2$    \\
  \hline $y_i$     &  $0$  & $1$ & $g+1$    & $2g+1$ & $2g+2$ \\
  \hline $f_i$     & $-i$  & $2$ & $2g+2-i$ &  $0$   & $0$    \\
  \hline $a_i$     &  $i$  & $0$ & $i$      &  $2$   & $2$    \\
  \hline $a_{g+1}$ &  $0$  & $1$ & $g+1$    & $2g+1$ & $2g+2$ \\
  \hline $a_{g+2}$ &  $1$  & $1$ & $g+2$    & $2g+3$ & $2g+4$ \\
  \hline $a_{2g}$  & $g-1$ & $1$ & $2g$     & $2g+1$ & $2g+2$ \\
  \hline $b_i$     &$1+i-g$& $1$ & $2+i$    &  $3$   & $4$    \\
  \hline
 \end{tabular}
\end{center}
 \caption{Weights and Measures}\label{fig:wam}
\end{figure}

Hereafter, we shall denote by $a_n$ ($1\leq n \leq g$),
as in (\ref{eq:adefn})
the positive representation
of the elementary symmetric polynomials in $n$-tuples of the $x_i$'s;
$b_i$ is as in (\ref{eq:bdefn}),
which is the corresponding coefficient in the curve given by
\begin{equation}\label{eq:balt}
\curve{D}:y=b_0 x^{g-1} + \cdots + b_{g-2} x + b_{g-1}
\end{equation}
which passes through each of the points $P_i$ and intersects the
curve $\curve{C}$ with the correct multiplicity at each of those
points;
we also define `extra' symmetric polynomials as follows:
\begin{equation*}\label{eq:extrasym}
 a_i=
 \begin{cases}
  y_1+y_2+\cdots+y_g & \text{when }i=g+1\text{,}\\
  x_1y_1+x_2y_2+\cdots+x_gy_g & \text{when }i=g+2\text{,}\\
  x_1x_2\ldots x_{g-1}y_g + \cdots + x_2x_3\ldots x_gy_1 & \text{for }i=2g\text{.}
 \end{cases}
\end{equation*}

\begin{conj} 
Given a hyperelliptic curve, $\curve{C}:y^2=F(x)$, of genus $g$, the
$a_i$ form a basis for the space spanned by the regular symmetric
polynomials in the pairs $(x_i,y_i)$, $(1\leq i\leq g)$, modulo the
relation defined by the underlying curve.
\end{conj}

The principal limitation at this point is that
currently the algorithms I have implemented in \texttt{maple} take a long
time to run, and sometimes the objects become too large for \texttt{maple}
to cope with. It is necessary to optimize to more appropriately balance
memory and CPU time considerations.

\section{Distillation}\label{sec:distillation}

One first decides on $n$ for the $\nThetag{n}$ embedding. Then choose a
$y$-weight and a $w$-weight to search. Collect all appropriate
monomial expressions in the $a$'s, $b$'s and $f$'s and separate into
levels, such that two expressions are in the same level iff they have the
same $o$-value; i.e., the order of the pole at $\infty$ is the same for
both. Then, using simple linear algebra, it is possible to find
linear combinations that sit in a lower level (i.e., the order of the pole
at $\infty$ is strictly less). Iterating from the highest
level to the lowest level is the most efficient way of determining
which combinations have a pole of order at most $n$. We give
an overview in the following diagram.

\begin{figure}
 \begin{center}
  \pstree{\XX{distillation}}{%
   \XX{get\_monos}\XX{init\_distil}\XX{next\_distil\_level}%
    \pstree{\XX{loop}}{\XX{process\_lists}\XX{get\_null\_space}%
     \XX{next\_distil\_level}}}
 \end{center}
 \caption{Top Down Analysis of \textsf{distillation}}
\end{figure}

\begin{figure}
\begin{center}
\pstree{\XX{get\_monos}}{\XX{get\_weights}%
 \pstree{\XX{wpartition}}{%
  \XX{init\_ctr}%
  \pstree{\XX{loop}}{\XX{checkweights}\XX{ctr\_inc}}}%
 \XX{translate}}
\end{center}
\caption{Top Down Analysis of \textsf{get\_monos}}
\end{figure}

The only problem is when $n$ is odd; for, when $d=2g+2$, $\nThetag{n}$
is empty, so this complicates finding a basis for $\nThetag{n}$ which
is independent of the degree $d$ of the curve $\curve{C}$. In this case,
we must initially pretend that $d$ is odd (i.e. $d'=d-1=2g+1$) to do an
initial distillation to obtain expression list $1$.
Then, a further distillation for $n$ with $d=2g+2$, followed by
throwing away any expression in the new list for which the coefficient
of $f_d$ is zero to obtain list $2$.
Finally, mixing together list $1$ and list $2$ and performing a final
distillation on the resulting list gives the required output.

\section{Simplified Derivation in Genus 2}\label{sec:simpgen2}

Recall that in genus~2 our curve~$\C$ is
as in (\ref{eq:genus2}) and a member of the Jacobian
can be described as $(x_1,y_1) + (x_2,y_2) - \infty^+ - \infty^-$.
Then the $a_i$, $b_i$, $\delta$ of
(\ref{eq:adefn}), (\ref{eq:bdefn}), (\ref{eq:delta}), respectively, are

\begin{equation}\label{eq:gen2abd}
\begin{array}{rcl}
&a_0 = 1,\ a_1 = x_1 + x_2,\ a_2 = x_1 x_2,\\
&b_0 = (y_2 - y_1)/(x_2 - x_1),\ b_1 = (x_2 y_1 - x_1 y_2)/(x_2 - x_1),\\
&\delta = x_2 - x_1.\\
\end{array}
\end{equation}

As we have already observed, $a_0,a_1,a_2$ all have
$\wt(a_i) \leq 2$ and so are all in $L(2\Theta)$. 
We are therefore missing one basis element.
However, $\wt(b_0) = \wt(b_1) = 4$, and do not give the
missing basis member. The natural next stage to investigate
quadratic monomials, such as

\begin{equation}\label{eq:b0sq}
b_0^2 = \bigl( F(x_1) + F(x_2) - 2 y_1 y_2 \bigr) / (x_1 - x_2)^2.
\end{equation}

Now, $\wt (b_0^2) = 8$, which seems even worse than before.
However, note that $\wt( (x_1 - x_2)^2 ) = 4$, and so
we only require that all terms in the numerator have $\wt$
at most~$6$, a condition already satisfied by the $y_1 y_2$ term.
Available to us is the fact that we can adjust $b_0^2$ by
polynomials in $a_0,a_1,a_2$, equivalent to adjusting
$F(x_1) + F(x_2)$ modulo $\delta^2$.
Now note, for example, that the terms of highest $\wt$
are $f_6 x_1^6 + f_6 x_2^6$, which satisfy $\wt (f_6 x_1^6 + f_6 x_2^6) = 12$.
But subtracting $f_6 (x_1 + x_2)^4(\delta^2)$ immediately reduces this~12
to~10. This motivates the following inductive procedure.

\begin{define}\label{def:reduc}
Let $p = p(x_1,x_2)$ be a symmetric polynomial in $x_1,x_2$,
with highest weight terms $c(x_1^k x_2^\ell + x_1^\ell x_2^k)$
where $c$ is a constant and,
without loss of generality, we take $k \geq \ell$.
If $k \geq \ell + 2$ then replace $p$ by
\begin{equation}\label{eq:reduc}
p - c x_1^\ell x_2^\ell ( x_1 + x_2 )^{k - \ell - 2} \delta^2.
\end{equation}
Continue this process until it is no longer the case
that $k \geq \ell + 2$ for the highest weight terms.
We denote the final result as $\hbox{Red}_\wt (p)$ modulo $\delta^2$.
\end{define}

We can now more succinctly describe $\ka_3$ of (\ref{eq:gen2kum})
as: $\ka_3 = \hbox{Red}_\wt (b_0^2)$ modulo $\delta^2$.
As a polynomial in $a_0,a_1,a_2,b_0,b_1$ this is

\begin{equation}\label{eq:ka4}
\ka_3 = b_0^2 - f_2 - f_3 a_1 - f_4 a_1^2
  + f_5 a_1 a_2 - f_5 a_1^3 - f_6 a_2^2 + 2 f_6 a_2 a_1^2 - f_6 a_1^4.
\end{equation}

One can check directly that $x_1^3$ and $x_2^3$ are the
highest powers of $x_1$ and $x_2$ in the numerator of
the right hand side, so that $\wt (\hbox{numerator}) = 6$
and $\wt (\ka_3 ) = 2$, as required.

For the 16 basis elements of $L(4\Theta)$, one cannot
simply apply the above process to monomials
in $a_0,a_1,a_2,b_0,b_1$, and it is necessary to
consider more general polynomials. For example,

\begin{equation}\label{eq:z7}
z_7 = -a_2 b_0 - a_1 b_1.
\end{equation}

Similarly, all of $z_1,\ldots , z_{15}$ in (\ref{eq:gen2z})
can be written as polynomials in $a_0$, $a_1$, $a_2$,
$b_0$, $b_1$. 

\section{$L(2\Theta)$ in Genus 3}\label{sec:L2Tgen3}

In genus~3, we find that (\ref{eq:adefn}),(\ref{eq:bdefn}),(\ref{eq:delta})
are

\begin{equation}\label{eq:gen3abd}
\begin{array}{rcl}
&a_0 = 1,\ a_1 = x_1 + x_2 + x_3,\\
&a_2 = x_1 x_2 + x_2 x_3 + x_3 x_1,\ a_3 = x_1 x_2 x_3,\\
&b_0 =
  (x_1 y_2 - x_2 y_1 - x_3 y_2 + x_3 y_1 - x_1 y_3 + x_2 y_3)/\delta,\\
&b_1 =
  (x_3^2 y_2-x_3^2 y_1+x_2^2 y_1+y_3 x_1^2-y_2 x_1^2-y_3 x_2^2)
  /\delta,\\
&b_2 =
(-x_2^2 x_3 y_1 + x_2^2 x_1 y_3 + x_3^2 x_2 y_1-
 y_3 x_2 x_1^2 + y_2 x_3 x_1^2 - x_3^2 x_1 y_2)/\delta,\\
&\delta = (x_1-x_2)(x_2-x_3)(x_3-x_1).\\
\end{array}
\end{equation} 

In genus~3, we find that, even for $L(2\Theta)$ it is not
enough merely to consider monomials in the $a_i,b_j$.
There is only one case, namely that of $b_0^2$ for which the
computations are similar to $b_0^2$ in genus~2. Namely,
one adjusts the numerator modulo~$\delta^2$ to obtain
a member of $L(2\Theta)$ in the form $b_0^2$ added
to a polynomial in $a_0,\ldots ,a_3$.  

\begin{equation}\label{eq:c0}
c_0 =  a_0b_0^2-f_7a_1^3+f_7a_2a_1-f_6a_1^2+3f_7a_3+2f_6a_2.
\end{equation}

However, we know from Lemma~\ref{lem:dimLnT} that
the dimension of $L(2\Theta)$ is $2^g = 8$, and
$a_0,a_1,a_2,a_3,c_0$ are five independent elements.
We are therefore missing three further independent elements
to make up a basis, and hence a set of coordinates
for the Kummer variety. We search amongst linear combinations
of monomials of the form $a_i b_j b_k$ to obtain expressions
that can be weight reduced to give new independent members
of $L(2\Theta)$. The expressions quickly become large,
and so it is helpful to have the following  
technical lemma.

\begin{lemma}\label{lem:wtslnt}
Let $\hbox{wt}$ be a weight for which $\C : y^2 = f_n x^n + \ldots + f_0$
and $\delta$ are homogeneous. Let $c$ be a member of $L(n\Theta)$
with denominator a power of~$\delta$. Let $c_{i_1}, \ldots ,c_{i_m}$
be the homogeneous pieces of~$c$ satisfying $c = c_{i_1}+ \ldots +c_{i_m}$
and $\hbox{wt}(c_{i_1}) = i_1, \ldots , \hbox{wt}(c_{i_m}) = i_m$,
where $i_1,\ldots ,i_m \in \bbbz$ are distinct.
Then each of $c_{i_1}, \ldots ,c_{i_m}$ is a member of
$L(n\Theta)$.
\end{lemma}

\begin{proof}
Let $t$ be any scalar; that is, $t$ is a member of the groundfield~$K$ 
which, as always, we assume to be of characteristic~0.
Replace every occurrence of $x,y,x_j,y_k,f_\ell$ 
by $t^{wt(x)}x, t^{wt(y)}y, t^{wt(x)}x_j,
t^{wt(y)}y_k, t^{wt(f_\ell)}f_\ell$, respectively.
Then the curve $\C$ is unchanged, and so the corresponding
$c(t)$ will still be in $L(n\Theta)$ for any value of~$t$.
But $c(t) = t^{i_1} c_{i_1} + \ldots + t^{i_m} c_{i_m}$.
We can now use the inverse of the invertible matrix
$( j^{i_k} )_{jk}$ to express each of $c_{i_1},\ldots , c_{i_m}$
as a linear combination of $c(1),\ldots ,c(m)$. Hence,
each $c_{i_1},\ldots , c_{i_m}$ is in $L(n\Theta)$, since membership
of $L(n\Theta)$ is preserved by linear combinations.
\end{proof}

The above lemma provides an invaluable computational aid, since
it tells us that there exists a basis for $L(n\Theta)$, where each
basis element is homogeneous with respect to any weight satisfying
the conditions of the lemma. Note that these conditions
are not met by $\wt$ itself, but they are satisfied, for
example, by the $x$ and $y$ weights in Figure~\ref{fig:wam}.
This means that when we search for linear combinations of
monomials, we can first partition them into sets of
equal $x$ and $y$ weights.

Taking the various squares and cross-products we notice that the
coefficients of $y_iy_j$ in the numerator are still too `heavy'
compared to the
denominator, and so we try to find linear combinations of $a_ib_jb_k$
which have a $y_1y_2$ term (in the numerator) of weight 10 or less;
then by symmetry we can
concentrate solely on reducing the weight of the rest of the terms in
these linear combinations.

The process is made more efficient by, for each product,
considering the $y_1y_2$
coefficient, separating the heaviest portion (in this case weight$>
10$) and symmetrising\footnote{Not as trivial as it sounds;
implemented in \maple{gpolyutil[get\_orbit]()}}
this portion\footnote{Separation \& re-symmetrification performed
by \maple{gpolyutil[get\_weight\_greaterthan]()}}. Then obtaining a
basis for the
null space of the set of the symmetric heavy portions gives the required
linear combinations of $a_ib_jb_k$ which have acceptably heavy (or
`light') $y_iy_j$ terms.

We find that the required linear combinations of the $a_ib_jb_k$ are

\begin{equation}\label{eq:kuminit}
\begin{array}{l}
\gamma_0 = a_0b_0^2\\
\gamma_1 = a_1b_0^2 + 2 a_0b_0b_1\\
\gamma_2 = a_0b_1^2 - a_2b_0^2\\
\gamma_3 = a_1b_1^2 + 2a_2b_0b_1 + a_3b_0^2
\end{array}
\end{equation}

Note however that the $\gamma_i$ are still too heavy, but after weight
reduction\footnote{\maple{gpolyutil[naive\_reduce\_g3()]}}
we find $\wred \gamma_i\in\embed{2}$ as required.

Using these ideas, one
can determine the $\nThetag{2}$ embedding for a hyperelliptic
curve of genus~$g=3$.

\begin{theorem}\label{thm:g3kum}
Let $a_0,a_1,a_2,a_3,b_0,b_1,b_2$ be as in (\ref{eq:gen3abd}).
A basis for $L(2\Theta)$, and hence an embedding
in $\mathbb{P}^8$ of the Kummer variety,
is given by:
\[
 [\ka_0,\ka_1,\ka_2,\ka_3,\ka_4,\ka_5,\ka_6,\ka_7] = [a_0,a_1,a_2,a_3,c_0,c_1,c_2,c_3]
\]
where
\begin{equation}\label{eq:g3kum}
\begin{align*}
c_0 = \;& a_0b_0^2-f_7a_1^3+f_7a_2a_1-f_6a_1^2+3f_7a_3+2f_6a_2\\
c_1 = \;& a_1b_0^2+2a_0b_0b_1-f_7a_1^4+3f_7a_2a_1^2-f_6a_1^3-f_7a_2^2\\
      &-f_7a_3a_1+2f_6a_2a_1-f_5a_1^2+2f_5a_2\\
c_2 = \;& a_0b_1^2-a_2b_0^2+f_7a_2a_1^3-2f_7a_2^2a_1+f_6a_2a_1^2\\
      &+f_7a_3a_2-f_6a_2^2+f_5a_2a_1-3f_5a_3\\
c_3 = \;& a_1b_1^2+2a_2b_0b_1+a_3b_0^2\\
      &+f_7a_2^2a_1^2-f_7a_3a_1^3+f_7a_3a_2a_1-f_7a_2^3+f_6a_2^2a_1\\
      &-f_6a_3a_1^2+f_5a_2^2-f_5a_3a_1\text{.}
\end{align*}
\end{equation}
\end{theorem}

\begin{proof}
We know from Lemma~\ref{lem:dimLnT} that the dimension
of $L(2\Theta)$ is~$8$. Furthermore it may be directly checked
algebraically that the above are all
linearly independent, for example by examining the
coefficient of $y_i y_j$ in the numerators of
$c_0,c_1,c_2,c_3$, and so $a_0,a_1,a_2,a_3,c_0,c_1,c_2,c_3$ are a basis.
It is sufficient to demonstrate the independence of the $c_i$ for
$i=0,\ldots ,3$. Considering the coefficient of
$y_1y_2(x_1-x_3)(x_2-x_3)$ in the numerator of $c_i$ gives
\begin{equation}\label{eq:cicoeff}
\begin{array}{l}
    c_0 \longmapsto -2\\
    c_1 \longmapsto 2 x_3\\
    c_2 \longmapsto -2 x_3^2\\
    c_3 \longmapsto -2 x_3^3\\
\end{array}
\end{equation}
which shows that the $c_i$ are independent as required.
\end{proof}

We have deliberately avoided mention of the case
where the genus is odd and $y^2 = F(x)$, where the
degree of $F(x)$ is even. In genus~3, this means that
we have considered only $F(x)$ of degree~$7$.
However, it is worth noting that one can formally apply
precisely the same procedures as above to the case where
$F(x)$ has degree~$8$. In that case, $\wt (x) = 2$, $\wt (y) = 8$,
and we can take the weight reductions of precisely the
same expressions as in~(\ref{eq:kuminit}).
The resulting revised versions of $c_0,\ldots c_4$
(which will include terms with
a coefficient of~$f_8$) still have only $\wt = 2$.
The problem here (if one wants to retain objects defined
over the ground field) is giving an interpretation of the
$x_1, x_2, x_3$, $y_1,y_2,y_3$ in terms of divisor classes,
since each divisor class is no longer canonically represented
by three points. The problem in representing divisor classes
by sets of points is that there is no equivalent  
of $g\Theta$ (when the degree of~$F(x)$ is odd)
or $(g/2)\Theta^+ + (g/2)\Theta^-$ (when the genus is~even).
We can pick one of $\Theta^+$ or $\Theta^-$, but neither
is defined over the ground field. However, as $-1$ acts trivially on
the Kummer variety and $c_0,\ldots,c_3$ furnish an embedding of this,
it may be
possible to extend an 
interpretation of $c_0,\ldots ,c_3$ to the degree~$8$ case,
although we have not been able to do so. Note that in
genus~$4$ there would be no such problems, since
the genus is even, and so the above techniques (like in genus~2) would
apply equally well to the cases where $F(x)$ has even or odd degree.

\section{Calculating the $c_i$}\label{sec:calcci}

The $c_i$ may be generated easily using the software in the
\texttt{hell} subdirectory of the ftp site. To read in to
\maple{maple} all of the files necessary, use \maple{thesis.v5}; then
the $c_i$ may be deduced from the output of the function
\begin{equation}
\text{\maple{get\_nTheta(genus,wWeight,yWeight,nTheta)}}
\end{equation}
as follows:
\begin{equation}
 \begin{array}{l}
  \maple{get\_nTheta(3,4,2,2)}\longmapsto c_0\\
  \maple{get\_nTheta(3,5,2,2)}\longmapsto c_1\\
  \maple{get\_nTheta(3,6,2,2)}\longmapsto c_2\\
  \maple{get\_nTheta(3,7,2,2)}\longmapsto c_3\text{.}
 \end{array}
\end{equation}

The function returns an array, which maps order of the pole on
$\Theta$ to the function; hence, the appropriate functions are
returned in the \maple{[2]} element. Also, note that this is a list of
polynomials in the $a_i,b_i$, from which an appropriate can be
chosen. Further, note that the function uses a triple pass of the
process of distillation (\ref{sec:distillation})
which returns results which are {\it formally
valid}, in the sense that they are simply polynomials that
match the weight parameters given without regard to whether the
results make sense (particularly with respect to the odd genus/even
degree problem described above); hence, the results will contain
references to $f_8$, although if the assignment $f_8=0$ is
subsequently made, the results can be chosen from the lists offered to
be identical to the ones given in (\ref{eq:g3kum}).

\section{$L(3\Theta)$ in Genus 3}\label{sec:L3Tgen3}

For $L(3\Theta)$ in genus~3, we know that the dimension is~27.
Furthermore, $a_0$, $a_1$, $a_2$, $a_3$, $b_0$, $b_1$, $b_2$
of~(\ref{eq:gen3abd})
are all in $L(3\Theta)$, as are the $c_0,c_1,c_2,c_3$
of Theorem~\ref{thm:g3kum}. This gives~11 independent members
of $L(3\Theta)$, and so 16 additional independent members
are required to give a basis, and hence an
embedding of the Jacobian as a projective variety.
Our strategy for finding these
is the same as for finding $c_0,c_1,c_2,c_3$; however, problems
with the sheer size of the algebra involved has prevented the
explicit determination of an entire basis. Nevertheless, we have
made some progress towards a basis, as described in the following
partial result.

\begin{theorem}\label{thm:g3L3}
Let $a_0,a_1,a_2,a_3,b_0,b_1,b_2$ be as in (\ref{eq:gen3abd}),
and $c_0,c_1,c_2,c_3$
be as in Theorem~\ref{thm:g3kum}. Then these, together with
the following, give~15 independent members of the
27-dimensional space $L(3\Theta)$.
\begin{equation}\label{eq:g3L3}
\begin{align*}
d_0 =\; &a_{{1}}{b_{{0}}}^{2}+a_{{0}}b_{{0}}b_{{1}}\\
     &+2\,f_{{5}}{a_{{0}}}^{3}a_{{2}}-f_{{5}}{a_{{0}}}^{2}{a_{{1}}}^{2}
      +3\,f_{{6}}{a_{{0}}}^{3}a_{{3}}+f_{{6}}{a_{{0}}}^{2}a_{{1}}a_{{2}}\\
     &-f_{{6}}a_{{0}}{a_{{1}}}^{3}-f_{{7}}{a_{{0}}}^{2}{a_{{2}}}^{2}
      +2\,f_{{7}}a_{{0}}{a_{{1}}}^{2}a_{{2}}-f_{{7}}{a_{{1}}}^{4},\\
d_1 =\; &a_0 b_0 b_2
      -f_6 a_0^2 a_1 a_3 - f_7 a_0 a_1^2 a_3,\\
d_2 =\; &a_0 b_1 b_2 + a_3 b_0^2\\
      &-f_{{5}}{a_{{0}}}^{2}a_{{1}}a_{{3}}+f_{{6}}{a_{{0}}}^{2}a_{{2}}a_{{3}}
       -f_{{6}}a_{{0}}{a_{{1}}}^{2}a_{{3}}+2\,f_{{7}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}
       -f_{{7}}{a_{{1}}}^{3}a_{{3}},\\
d_3 =\; &a_{{1}}b_{{1}}b_{{2}}+a_{{2}}b_{{0}}b_{{2}}-a_{{3}}b_{{0}}b_{{1}}\\
      &-f_{{5}}{a_{{0}}}^{2}a_{{2}}a_{{3}}-f_{{6}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}
      +f_{{7}}a_{{0}}{a_{{2}}}^{2}a_{{3}}-f_{{7}}{a_{{1}}}^{2}a_{{2}}a_{{3}}.\\
\end{align*}
\end{equation}
\end{theorem}

\begin{proof}
It is sufficient to demonstrate the independence of the $c_i,d_i$ for
$i=0,\ldots ,3$. Considering the coefficient of
$y_1y_2(x_1-x_3)(x_2-x_3)$ in the numerator of the $c_i,d_i$ extends~(\ref{eq:cicoeff})
as follows.
\begin{equation}
  \begin{array}{l}
    c_0 \longmapsto -2\\
    c_1 \longmapsto 2 x_3\\
    c_2 \longmapsto -2 x_3^2\\
    c_3 \longmapsto -2 x_3^3\\
    d_0 \longmapsto -(x_1 + x_2)\\
    d_1 \longmapsto -(x_1 + x_2) x_3\\
    d_2 \longmapsto (x_1 + x_2) x_3^2\\
    d_3 \longmapsto (x_1 + x_2) x_3^3\\
  \end{array}
\end{equation}
which shows that the $c_i,d_i$ are independent as required.
\end{proof}

Note that, by inspection, the $c_i, d_i$ are homogeneous
in the $x$, $y$ and $w$ weights of Figure~\ref{fig:wam}.
We can take the case $g=3$, $n=7$ of Figure~\ref{fig:wam}
and extend the weights to the $c_i$ of~(\ref{eq:g3kum})
and $d_i$ of~(\ref{eq:g3L3}),
as follows.

\begin{figure}[h]
\begin{center}
 \begin{tabular}{|r||c|c|c|c|}
  \hline           &  $x$  & $y$ & $w$   & $o_{7}$ \\
  \hline $x_i$     &  $1$  & $0$ & $1$   &  $2$ \\
  \hline $y_i$     &  $0$  & $1$ & $4$   & $7$ \\
  \hline $f_i$     & $-i$  & $2$ & $8-i$ &  $0$   \\
  \hline $a_i$     &  $i$  & $0$ & $i$   &  $2$  \\
  \hline $a_{4}$   &  $0$  & $1$ & $4$   & $7$  \\
  \hline $a_{5}$   &  $1$  & $1$ & $5$   & $9$ \\
  \hline $a_{6}$   & $2$   & $1$ & $6$   & $9$ \\
  \hline $b_i$     & $i-2$ & $1$ & $2+i$ &  $3$  \\
  \hline $c_i$     & $i-4$ & $2$ & $4+i$ &  $2$  \\
  \hline $d_i$     & $i-3$ & $2$ & $5+i$ &  $3$  \\
  \hline
 \end{tabular}
\end{center}
 \caption{Weights and Measures}\label{fig:wamext}
\end{figure}

We presume that to find the missing~12 members of the basis
of~$L(3\Theta)$ would require a search through monomials
cubic in the~$b_i$. At the moment there are simply too
many of these to perform an exhaustive search, even
with the enhancements described above.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \input algolnt.tex

\chapter{Algorithms for Determining $\mathcal{L}(n\Theta)$}
\label{ch:algolnt}
%programs for adding divisors on hyperelliptic curves
%Methods for finding L(n*Theta). Pogram descriptions
%in detail and Algorithms, worked examples.

The aim of this chapter is to give detail and
precision to the descriptions of the algorithms motivated
by the previous sections.

\section{Finding Monomials with a Given Weight}

To determine the list of the monomials of combinations of the symbols
$f_i$, $a_i$, $b_i$, at a particular $w$-weight, $w$, and $y$-weight,
$y$, we require a \emph{weighted partition} of $w$ and $y$. For this
purpose, we need a list of weighted counters, $[c_1,\ldots,c_n]$, such
that there is a $1:1$ correspondence between the $c_i$ and the $f_i$,
$a_i$ and $b_i$; and each $c_i$ has $w$-weight $w_i$ and $y$-weight
$y_i$. Hence the only permissible values of the counters are those
with $\sum c_iw_i = w$, and $\sum c_iy_i = y$. As it is possible that
$y_j=0$ for some specific $j$, it is simpler (and far quicker) to
perform both comparisons at the same time.

In this implementation of the function \texttt{wpartition}, the
expected arguments are in pairs; the first in the pair being the
maximum allowed weight sum for that measure, and the second being a
list of weights for the counters (which also gives the number of
counters required): e.g., to partition the number 3 into 2 parts, each
with weight 1: \texttt{wpartition(3,[1,1])} returns
\[[[3,0],[2,1],[1,2],[0,3]]\]
and to partition the number 3 into 2 parts with the second part being
weighted at 2: \texttt{wpartition(3,[1,2])} returns
\[[[3,0],[1,1]]\]

%wpartition
\begin{alltt}{\btt{}andy_util[ wpartition ] := proc()
  local flag, ctr_idx, ctr_cur, ctr_new, ctr_len, retval;

  if (nargs<2) then ERROR(`too few arguments`) fi;
  if type(nargs,odd) then ERROR(`wrong number of arguments`) fi;

  \ccomment{# retval  the value we return; this will be a list of a copy}
  \ccomment{#         of the counter list each time it meets the weight}
  \ccomment{#         criteria}
  \ccomment{# args[2] must be a list of weights; the list will contain the}
  \ccomment{#         same number of elements as the number of counters we}
  \ccomment{#         require so we init the current counter, ctr_cur with}
  \ccomment{#         zeros exactly to the right length.}
  \ccomment{# flag    will be set to true upon a counter overflow; at which}
  \ccomment{#         point we have ticked past the end of the counter}
  \ccomment{#         through all the possibilities, and may return.}
  retval:=[];
  ctr_len:=nops(args[2]);
  ctr_cur:=[0$ctr_len];
  flag:=false;

  while(flag=false) do
    \ccomment{# the function checkweights verifies whether the current}
    \ccomment{# counter meets all the criteria specified; the `e'}
    \ccomment{# argument refers to `equal to'; as the function can also}
    \ccomment{# check for `le', i.e., `less than or equal to'.}
    \ccomment{# If the criteria are met, it returns true, in which case}
    \ccomment{# the value of the current counters is appended to the}
    \ccomment{# return value.}
    if andy_util[checkweights](ctr_cur,"e",args)
      then retval:=[op(retval),ctr_cur] fi;
    \ccomment{# the ctr_inc function increments the current counter list}
    \ccomment{# on a `tick'; so that if incrementing counter c_i causes}
    \ccomment{# the weight to be too high for some weight specification,}
    \ccomment{# then c_1 to c_i are all reset to zero, and c_{i+1} is}
    \ccomment{# incremented. If i+1 is past the end of the list, then}
    \ccomment{# we have overflowed and wpartition can return.}
    \ccomment{# ctr_inc returns a list; [boolean,new_counter] where}
    \ccomment{# the boolean refers to whether or not an overflow occurred.}
    andy_util[ctr_inc](ctr_cur,args);
    flag:=%[1]; ctr_cur:=%%[2];
  od:

  retval;
end:}\end{alltt}

The function \texttt{ctr\_inc} is implemented as follows 

\begin{alltt}{\btt{}andy_util[ ctr_inc ] := proc( ctr::list )
  local ctr_len, ctr_idx, ctr_new, flag;

  \ccomment{# the arguments following the list provide the same information}
  \ccomment{# as given to wpartition above}

  ctr_len:=nops(ctr);
  \ccomment{# try to inc the least significant counter first}
  ctr_idx:=1;
  \ccomment{# assume that the counter will overflow on this inc}
  \ccomment{# this flag also doubles as being the logical inverse of}
  \ccomment{# whether the counter has been incremented successfully}
  flag:=true;
  \ccomment{# the condition below translates to: }
  \ccomment{#   whilst we have not incremented successfully, and have not}
  \ccomment{#   overflowed...}
  while(flag=true) and (ctr_idx<=ctr_len) do
    \ccomment{# form a new list in which the counters up to ctr_idx are}
    \ccomment{# all nullified, and the counter at ctr_idx is incremented}
    \ccomment{# by 1.}
    ctr_new:=[ 0$(ctr_idx-1), ctr[ctr_idx]+1,
               op(ctr[ctr_idx+1..ctr_len])];
    \ccomment{# if the increment is still too heavy, then we will need to}
    \ccomment{# repeat the operation again on the next counter}
    ctr_idx:=ctr_idx+1;
    if(andy_util[checkweights](ctr_new,"le",args[2..nargs]))
    then flag:=false fi;
  od:
  \ccomment{# at this point the flag is false only if we managed to}
  \ccomment{# increment the counter list with a weight within the}
  \ccomment{# boundaries specified in the arguments; otherwise there}
  \ccomment{# was an overflow and there are no more values possible}
  [flag,ctr_new];
end:}\end{alltt}

The function \texttt{checkweights} is used by both
\texttt{wpartition}, which requires a test for equality,
and \texttt{ctr\_inc}, which requires a test for
less than or equal to, and is implemented as follows:

%checkweights
\begin{alltt}{\btt{}andy_util[ checkweights ] := proc( check::list, rule::string )
  local i, value, weightlist;

  \ccomment{# the first two supplied arguments are the list that requires}
  \ccomment{# checking, and the rulestring, which determines whether the}
  \ccomment{# test is for equality, or for `less than or equals'.}
  \ccomment{# the following arguments are as supplied to wpartition; i.e.,}
  \ccomment{# total weight, and a list of each component weight}
  for i from 3 to nargs by 2 do
    \ccomment{# we cycle through all the weight/[component weight] values}
    \ccomment{# testing each time}
    value:=args[i];
    weightlist:=args[i+1];
    \ccomment{# n.b. there is a small efficiency question here, as these}
    \ccomment{# tests are made on every loop; it would be more efficient,}
    \ccomment{# although programmatically more messy to perform the test}
    \ccomment{# once and have two different loops}
    if ((rule="e") and
        (value<>sum('weightlist[i]*check[i]',
                    'i'=1..nops(check))))
      then RETURN(false) fi;
    if ((rule="le") and
        (value<sum('weightlist[i]*check[i]','i'=1..nops(check))))
      then RETURN(false) fi;
  od;
  \ccomment{# we return true iff every test passed successfully}
  true;
end:}\end{alltt}

Finally, to obtain the list of monomials in the symbols $f_i$, $a_i$,
and $b_i$, with $w$-weight $w$ and $y$-weight $y$, we use the function
\texttt{get\_monos}; e.g., \texttt{get\_monos(8,4,2)}, which will
return a list of monomials of $w$-weight $4$ and $y$-weight $2$ on a
hyperelliptic curve defined by an equation of degree $8$ in $x$. The
function is:

\begin{alltt}{\btt{}hell2[ get_monos ];= 
     proc( deg::integer, wweight::integer, yweight::integer )
  local genus, fablist, fabwwt, fabywt_hyper, i, j, exps;

  genus:=floor((deg-1)/2);
  \ccomment{# create a list of the f_i, a_i, and b_i; this will correspond}
  \ccomment{# to the counters used}
  fablist:=[ seq(f[i],i=0..deg),
             seq(a[i],i=1..2*genus),
             seq(b[i],i=0..genus-1)];
  \ccomment{# the w-weight of each of the symbols in the above list is}
  \ccomment{# placed in a list}
  fabwwt:=[seq(hell2[wwt](deg,fablist[i]),i=1..nops(fablist))];
  \ccomment{# the y-weight of each of the symbols in fablist is now}
  \ccomment{# placed in a list}
  fabywt_hyper:=
    [seq(hell2[ywt_hyper](deg,fablist[i]),i=1..nops(fablist))];

  \ccomment{# now calculate the weighted partition of the w-weight and}
  \ccomment{# y-weight given corresponding weights of the counters}
  \ccomment{# exps will be a list of weighted partitions satisfying the}
  \ccomment{# stated constraints}
  exps:=
  andy_util[wpartition]( wweight, fabwwt, 
                         yweight, fabywt_hyper );

  \ccomment{# for each partition in exp, create the monomial, and add it}
  \ccomment{# to a list, which becomes the return value of this function}
  [seq(product('fablist[i]^exps[j][i]',
                    'i'=1..nops(fablist)),j=1..nops(exps))];
end:
}\end{alltt}

\section{Weight Reduction}

As discussed earlier, to each symbol can be associated a
hyperelliptic weight. In particular, for a genus $g$ curve
$\mathcal{C}$ defined by a hyperelliptic equation of degree $d$ in
$x$, we can associate a weight, $2$, with $x$, and a weight $d$ with
$y$.  The function \texttt{g3weight} returns the hyperelliptic weight
of the symbols $x$, and $y$ for a genus 3 curve given by a degree 7
equation $F(x,y)=0$ with $F(x,y)=y^2-f_0+\cdots+f_7 x^7$. Note that
$F$ is pseudo-homogeneous with respect to this weight.

\begin{define}
Suppose $p$ is a rational symmetric function on $\mathcal{C}^3$,
defined in terms of $(x_i,y_i)$, for $1\leq i\leq 3$. Then the
\texttt{g3weight} of this function is defined by the weight of its
denominator subtracted from the weight of its numerator, the weights
of which are determined by calculating the degree of $p$ wrt $w$ when
$(x_1,y_1)$ is substituted by $(wx_1,wy_1)$.
\end{define}

Note that the use of $x_1,y_1$ as a `pivot' is a common technique in
these algorithms.

%g3weight
\begin{alltt}{\btt{}g3w(p((x1,y1),(x2,y2),(x3,y3)))=
       degree(p((w*x1,w*y1),(x2,y2),(x3,y3)),w)
g3weight(p)=g3w(numerator(p))-g3w(denominator(p))}\end{alltt}

Often in the process of `weight reduction' we wish to obtain the
heaviest portion, $h$, of an expression $p$. Again, we pivot on $x_1,y_1$ and
construct an expression $e$ by considering the weight of each term $t$
in $p$. If $t$ is heavier than $e$ then we discard the contents of $e$
and start again by setting \texttt{e:=t}; on the other hand, if $t$ is the
same weight as $e$ then we append $t$ to $e$, thus:
\texttt{e:=e+t}. Otherwise, no action is needed. The return from this
routine, shown below, is no longer symmetric, as just terms in $x_1,y_1$ are
considered, but a symmetric result can be obtained by using another
function, \texttt{get\_orbit}.

%get_heaviest_terms
\begin{alltt}{\btt{}get_heaviest_terms:=proc(p,g)
  local m, n, terms, poly;
  poly:=expand(x[1]*y[1]*p):
  terms:=0:
  for m from 1 to degree(poly,y[1]) do
    expand(coeff(collect(poly,y[1]),y[1]^m));
    n:=degree(%,x[1]);
    if n>0 then
      coeff(collect(expand(%%),x[1]),x[1]^n);
      if gpolyutil[g3weight](terms)=((2*g+1)*(m-1)+2*(n-1))
        then terms:=terms+x[1]^(n-1)*y[1]^(m-1)*% fi;
      if gpolyutil[g3weight](terms)<((2*g+1)*(m-1)+2*(n-1))
        then terms:=x[1]^(n-1)*y[1]^(m-1)*% fi;
    fi;
  od:
  expand(terms);
end:}\end{alltt}

Before \texttt{get\_orbit()}, we require a simple function
\texttt{permute\_vars(p,sigma)}
that
permutes the $(x_i,y_i)$. Essentially this function works by
substituting $x_i$ with $t_i$, and $y_i$ with $s_i$, before then
substituting $t_i$ with $x_{i\sigma}$, and $s_i$ with
$y_{i\sigma}$. Exactly how $\sigma$ is represented is implementation
dependent. For instance, in Maple V, it is useful to use the variable
length parameter capability, so that the number in the $(i+1)$-th
position in the parameter list represents the $j$ in the $x_j$
which $x_i$ will become.

Consider the operation \texttt{strip}, which strips from an expression
all terms which contain neither $x_1$ nor $y_1$. Then, given that the
original expression was symmetric in the $(x_i,y_i)$, the resultant
expression still contains enough information to reconstruct the
original: this reverse operation 
is the purpose of the function \texttt{get\_orbit(poly,genus)}.
In essence, this
function obtains a list of permutations of the terms in \texttt{p}
and, after discarding any repetitions, returns the sum of all the
terms in that list. For instance,

{\btt{}get\_orbit(x1+x2+x1*x2,3) = x1+x2+x3+x1*x2+x1*x3+x2*x3}

% get orbit
\begin{alltt}{\btt{}gpolyutil[get_orbit]:=proc(poly,genus)
  local perms, terms, newterms, i, j;
  perms:=combinat[permute](genus):
  if type(expand(poly),`+`)
    then terms:=[op(expand(poly))]
    else terms:=[poly] fi;
  newterms:={};
  for i from 1 to nops(terms) do
    newterms:={ op(newterms),
                seq(gpolyutil[permute_vars](terms[i],
                                            op(perms[j])),
                    j=1..nops(perms))};
  od;
  sum('newterms[i]','i'=1..nops(newterms));
end:}\end{alltt}

Often we are interested not just in terms of a particular weight, but
in all terms heavier than some specific weight. Clearly we need
another function, \texttt{get\_weight\_greaterthan}. This takes 3
arguments, specifying the desired weight, the expression to be
`skimmed', and the genus of the underlying curve. The function simply
needs to call \texttt{get\_heaviest\_terms} repeatedly, making a note of
which terms are heaviest, removing them, and finally returning the
heaviest terms in a single expression.

%get_weight_greaterthan
\begin{alltt}{\btt{}gpolyutil[get_weight_greaterthan]:=proc(wght::integer,
                                        p::polynom,
                                        genus::integer)
  local symp, tempval, retval;
  if (genus <> 3)
    then ERROR(`only genus 3 implemented - weight function`) fi;
  symp:=get_orbit(p,genus);
  tempval := 0;
  retval := 0;
  while (g3weight(symp) > wght) do
    tempval := get_orbit(get_heaviest_terms(symp,genus),genus);
    retval := retval + tempval;
    symp := simplify(symp-tempval);
  od;
  retval;
end:}\end{alltt}

\section{Distillation}\label{sec:distilalgo}

The key distillation algorithms, as used in section \ref{sec:calcci},
are described here. The first one, \texttt{distillation} takes 2
arguments; the degree of $F$, and a list of monomials. The algorithm
depends on tables of lists of the monomials in different formats (both
expanded into the $x_i,y_i$ representation and the smaller $a_i,b_i$
representation.

\begin{alltt}{\btt{}\ccomment{# uses a lot of Maple V R5 lexical scoping}
hell[ distillation ] := proc( deg::integer, monos::list )
  local init_distil, next_distil_level, get_top_portion,
	w_subs_hyper, process_lists, update_tables,
	distil_levels, w, ndl, emweight,
        sym_table, exp_table, exp_list, top_list, top_kbasis,
        orders, genus, i, m, mweight, wweight;

  \ccomment{#######################################################}
  \ccomment{#### subprocedures}

  \ccomment{# creates table of lists of monomials indexed by their order}
  init_distil := proc()
    local m, morder;
    sym_table:=table([]);
    for m in monos do
      if type(m,`+`)
        then morder := hell[order_hyper]( deg, m );
        else morder := hell[order_symbol]( deg, m );
      fi;
      if not member( [morder], [indices(sym_table)] )
      then sym_table[morder]:=[]; exp_table[morder]:=[]
      fi;
      sym_table[morder]:=[op(sym_table[morder]),m];
      exp_table[morder]:=[op(exp_table[morder]),infinity];
    od:
    distil_levels := {indices(sym_table)};
  end:

  next_distil_level := proc()
    local i, ndl;
    if (nops(distil_levels) = 0) then RETURN(-infinity) fi;
    jdl := max(seq(op(distil_levels[i]),
                   i=1..nops(distil_levels)));
    distil_levels := distil_levels minus {[ndl]};
    ndl;
  end:

  w_subs_hyper := proc( expr )
    local xyexpr;
    xyexpr := hell[subs_hyper]( deg, expr );
    subs( x[1]=w^orders[x]*x[1], y[1]=w^orders[y]*y[1], xyexpr );
    collect( expand(w*%), w );
  end:

  process_lists := proc()
    local new_exp_list, i;
    new_exp_list := [];
    for i from 1 to nops(exp_list) do
      if (exp_list[i]<>infinity)
      then new_exp_list := [op(new_exp_list),exp_list[i]]
      else new_exp_list := 
        [op(new_exp_list),
         w_subs_hyper(delta^emweight*sym_table[ndl][i])]
      fi;
    od;
    exp_list := new_exp_list;
    top_list := 
      [seq(coeff(exp_list[i],w^(ndl+1+emweight*orders[delta])),
           i=1..nops(exp_list))];
  end:

  update_tables := proc()
    local i, basis_elt, ab_expr, wxy_expr, wxy_ord;
    for basis_elt in top_kbasis do
      ab_expr  := sum('sym_table[ndl][i]*basis_elt[i]',
                      'i'=1..nops(exp_list));
      wxy_expr := sum('exp_list[i]*basis_elt[i]',
                      'i'=1..nops(exp_list));
      wxy_ord  := degree(collect(expand(wxy_expr),w),w)
                  -1-emweight*orders[delta];
      if not ( member([wxy_ord], [indices(sym_table)]) )
      then
        distil_levels:={op(distil_levels),[wxy_ord]};
        sym_table[wxy_ord]:=[];
        exp_table[wxy_ord]:=[]
      fi;
      sym_table[wxy_ord]:=[op(sym_table[wxy_ord]),ab_expr];
      exp_table[wxy_ord]:=[op(exp_table[wxy_ord]),wxy_expr];
    od;
    exp_table[ndl]:=[]; exp_list:=[]; \ccomment{# free up some space !!!!}
  end:

  \ccomment{#######################################################}
  \ccomment{#### main}
  \ccomment{# this is the entry point of the algorithm.}
  wweight := hell[wwt](deg, monos[1]);
  mweight := hell[mwt](deg, monos[1]);
\ccomment{#  monos := hell[get_monos]( deg, wweight, mweight );}
\ccomment{#  monos := [seq(monos[i]*delta^mweight,i=1..nops(monos))];}

  emweight := 2*ceil(mweight/2);

  genus := floor((deg-1)/2);
  orders[x]     := hell[order_hyper]( deg, x[1] );
  orders[y]     := hell[order_hyper]( deg, y[1] );
  orders[delta] := hell[order_hyper]( deg, delta );

  init_distil();

  ndl := next_distil_level();
  while(ndl<>-infinity) do
    printf("%d ",ndl);
    exp_list := exp_table[ndl];
    process_lists();
    top_kbasis := mvutil[get_null_space](top_list,
                                         [seq(x[i],i=1..genus),
                                          seq(y[i],i=1..genus)]);
    update_tables();
    ndl := next_distil_level();
  od;

  eval(sym_table);
end:}\end{alltt}

The following algorithm uses the \maple{distillation} function to
perform a triple-pass distillation in order to deal with both cases,
$F$ being of odd or even degree, and return a list of monomials of the
correct weight, and having the correct order on $\Theta$ (when this
makes sense; cf \ref{sec:calcci}).

\begin{alltt}{\btt{}hell[ get_nTheta ] := proc( genus::integer, wweight::integer,
                            mweight::integer, nTheta::integer)
  local odeg, edeg, i, j, m,
        emonos, eptable, omonos, optable, eolist,
        eTheta, oTheta, mature, theta_table, varlist;

  \ccomment{# given the genus, calculate both odd and even degrees}
  eTheta  := 2*ceil(nTheta/2);
  oTheta  := eTheta-1;
  edeg    := 2*genus+2;
  odeg    := edeg-1;

  \ccomment{# first distillation is for even degree}
  emonos  := hell[get_monos](edeg, wweight, mweight);
  eptable := hell[distillation](edeg, emonos);

  \ccomment{# second distillation is for odd degree}
  omonos  := hell[get_monos](odeg, wweight, mweight);
  optable := hell[distillation](odeg, omonos);

  \ccomment{# mix the tables; fist add the polynomials from the odd}
  \ccomment{# degree distillation}
  eolist  := [];
  for i in indices(optable) do
    if (op(i) <= eTheta)
      then eolist:=[op(eolist),op(optable[op(i)])];
    fi;
  od;

  \ccomment{# next, add those polynomials from the even degree}
  \ccomment{# distillation whose coefficient of f[edeg] is not 0}
  for i in indices(eptable) do
    for j from 1 to nops(eptable[op(i)]) do
      if coeff(collect(eptable[op(i)][j],f[edeg]),f[edeg])<>0
        then eolist := [op(eolist),eptable[op(i)][j]];
      fi;
    od;
  od;

  \ccomment{# final pass on distillation}
  mature := hell[distillation](edeg,eolist);

  theta_table := table([eTheta=[],oTheta=[]]);
  for m in mature[eTheta] do
    i := hell[order_hyper](odeg,m);
    if i<>(-infinity) and (hell[bwt](edeg,m)=mweight)
      then theta_table[i]:=[op(theta_table[i]),m];
    fi;
  od;

  varlist := [seq(a[i],i=1..2*genus),seq(b[i],i=0..genus-1),
              seq(f[i],i=0..edeg)];

  table([eTheta=mvutil[get_basis](theta_table[eTheta],varlist),
	 oTheta=mvutil[get_basis](theta_table[oTheta],varlist)]);
end:}\end{alltt}

\section{Overview of the Implementation of Distillation}

The implementation of the distillation algorithm,
$\maple{distillation}$, given above, makes a lot of use of a feature
of Maple V introduced in release 5; namely \emph{lexical scoping}. This
essentially allows procedures defined within other procedures access
to the local variables used by the parent procedure. As this can be
quite confusing, this section lists, in the style of technical
documentation, the various variables and sub-procedures and how they
interact. This will serve to allow for an example of the
implementation at work.

\subsection{Variables}

\noindent\begin{tabularx}{\linewidth}{|r|X|}
 \hline%
             $\maple{w}$ & used as a formal variable\\
 \hline%
       $\maple{mweight}$ & used only to calculate $\maple{emweight}$\\
 \hline%
      $\maple{emweight}$ & minimum power of $\delta$ by which one
                needs to multiply 
                 to generate
	         a list of symmetric polys, as opposed to symmetric ratpolys,
	         which may possibly have antisymmetric numerator and
                 denominator.\\
 \hline%
         $\maple{genus}$ & the genus of the curve\\
 \hline%
        $\maple{orders}$ & a list of orders of symbols,
                 indexed by the symbol; this is
	         just an optimisation\\
 \hline%
     $\maple{sym\_table}$ & table of lists of expressions
                 in $f$,$a$,$b$ indexed by its level;
	         i.e., $o$, the order of the pole at $\infty$
                 of that expression\\
 \hline%
     $\maple{exp\_table}$ & copy of $\maple{sym\_table}$
                 with each expression multiplied by
	         $\delta^\chi$, where $\chi=\maple{emweight}$, and expressed
                 in terms of $f$,$x$,$y$,$w$
	         where the power of $w$ in any term is equal to 1 plus 
	         the order of the
	         pole at $\infty$ of its coefficient. Expanding a
	         given expression is deferred until the expression is required;
	         until then, its place is marked with
	         the symbol $\infty$. When a level has been completed,
	         the expanded expressions for that level are no longer
                 required.\\
 \hline%
	   $\maple{ndl}$ & the index into $\maple{sym\_table}$
                 for the main loop of the algorithm; the
	         current level that is being processed.\\
 \hline%
      $\maple{exp\_list}$ & the fully expanded list from
                 $\maple{exp\_table}$ corresponding to $\maple{ndl}$\\
 \hline%
      $\maple{top\_list}$ & a list corresponding to $\maple{exp\_list}$,
                 but with each entry
	         truncated to include only the coefficient of the leading
                 term of $\maple{w}$.\\
 \hline%
    $\maple{top\_kbasis}$ & the set of vectors which form the basis for
                 the null space
	         of the polynomials in $\maple{top\_list}$\\
 \hline%
 $\maple{distil\_levels}$ & the set of levels yet to be processed\\
 \hline%
\end{tabularx}

\subsection{Subprocedures}

\noindent\begin{tabularx}{\linewidth}{|r|X|}
 \hline\multicolumn{2}{|c|}{$\maple{init\_distil}$}\\
 \hline\multicolumn{2}{|c|}{Initialize the main data 
  structures for the algorithm}\\
 \hline\multicolumn{2}{|l|}{\textbf{Variables used}}\\
 \hline$\maple{monos}$ & read-only\\
 $\maple{sym\_table}$ & initialize table of lists of expressions indexed by
  level\\
 $\maple{exp\_table}$ & initialize table by placing marker in every position\\
 $\maple{distil\_levels}$ & initialize with the keys from
  $\maple{sym\_table}$\\
 \hline
\end{tabularx}

\noindent\begin{tabularx}{\linewidth}{|r|X|}
 \hline\multicolumn{2}{|c|}{$\maple{next\_distil\_level}$}\\
 \hline\multicolumn{2}{|X|}{Determine next level to process. This function
  returns the highest entry from $\maple{distil\_levels}$, or
  $-\infty$ if empty.}\\
 \hline\multicolumn{2}{|l|}{\textbf{Variables used}}\\
\hline%
 $\maple{distil\_levels}$ & the return value is removed from the list\\
 \hline
\end{tabularx}

\noindent\begin{tabularx}{\linewidth}{|r|X|}
 \hline\multicolumn{2}{|c|}{$\maple{w\_subs\_hyper}$}\\
 \hline\multicolumn{2}{|X|}{%
  Given an expression, $\maple{expr}$, in terms of $a$, $b$, expand into
  expressions in terms of $x$, $y$, and introduce a formal variable,
  $w$, such that the coefficient of the term $w^{n+1}$ is a symmetric
  polynomial in the pairs $(x_i,y_i)$ with pole of order $n$ at $\infty$.
  The expanded expression is the return value.}\\
 \hline\multicolumn{2}{|l|}{\textbf{Parameters}}\\
\hline%
 $\maple{expr}$ & the expression to expand\\
 \hline
\end{tabularx}

\noindent\begin{tabularx}{\linewidth}{|r|X|}
 \hline\multicolumn{2}{|c|}{$\maple{process\_lists}$}\\
 \hline\multicolumn{2}{|X|}{%
  Process $\maple{exp\_list}$, replacing any $\infty$ markers with expanded
  expression, muliplying each expression by $\delta^\chi$ (where 
  $\chi=\maple{emweight}$) to remove all denominators, and ensure all
  expressions are symmetric, rather than anti-symmetric, in the pairs
  $(x_i,y_i)$. N.B., that these expressions will be polynomials
  in the variable $w$. The variable $\maple{top\_list}$ is generated
  by considering the coefficients of the leading terms in $w$ of each of
  the members of $\maple{exp\_list}$.
  }\\
 \hline\multicolumn{2}{|l|}{\textbf{Variables used}}\\
\hline%
 $\maple{emweight}$ & read-only\\
 $\maple{exp\_list}$ & list of expanded entries\\
 $\maple{top\_list}$ & list of coefficients of leading term in $w$ in
  corresponding expressions in the $\maple{exp\_list}$.\\
 \hline
\end{tabularx}

\noindent\begin{tabularx}{\linewidth}{|r|X|}
 \hline\multicolumn{2}{|c|}{$\maple{update\_tables}$}\\
 \hline\multicolumn{2}{|X|}{
  New expressions are generated by calculating linear combinations of
  the members of $\maple{sym\_table}$ and $\maple{expr\_table}$ as
  indicated by the members of $\maple{top\_kbasis}$. The pole at $\infty$
  of each of the resulting expressions is strictly less than those of the
  original expressions.
  The new expressions are added to the tables $\maple{sym\_table}$ and
  $\maple{exp\_table}$, and $\maple{distil\_levels}$ is updated if
  necessary.}\\
 \hline\multicolumn{2}{|l|}{\textbf{Variables used}}\\
\hline%
 $\maple{top\_kbasis}$ & vector list containing basis for null space of
  $\maple{top\_list}$.\\
 $\maple{sym\_table}$ & new entries are added, as determined by 
  $\maple{top\_kbasis}$.\\
 $\maple{exp\_table}$ & new entries are added, in a similar fashion to
  $\maple{sym\_table}$; old entries which are not needed again are discarded.\\
 $\maple{exp\_list}$ & used, along with $\maple{top\_kbasis}$ to generate
  new entries in $\maple{exp\_table}$; its value is discarded before return.\\
 $\maple{distil\_levels}$ & as a result of adding new entries to the tables,
  it might be necessary to process more levels; they are added if necessary.\\
 \hline
\end{tabularx}

\section{Example of Distillation}

As an example of the behaviour of my implementation of the
distillation algorithm, we now consider how it calculates $c_1$ in
genus $3$, as given in Theorem~\ref{thm:g3kum}. This is a good
representative of a nontrivial distillation in which the expressions
are still vaguely manageable by hand.

We know that
$c_1$ has $w$-weight $5$, and $y$-weight $2$, but if we had not known
these values, we could have arrived at them by deciding to fix on a
low value for the $y$-weight, which essentially limits the highest
power of the $b_i$'s in the expressions, and then searching through
the $w$-weights from the minimum possible for the given
$y$-weight.

Choosing the $y$-weight to be $2$ limits the expressions
to be quadratic in the $b_i$'s, and, given that the lightest (in terms
of the $w$-weight) expression that contains a quadratic $b_i$ is
$b_0^2$, which has weight $4$, this is a sensible place to
start. Choosing $4$ gives $c_0$; here we choose $5$ to obtain $c_1$.

We use the function $\maple{get\_monos(7,5,2)}$ to obtain a comprehensive
list of suitable monomials in $f$,$a$,$b$:
\begin{equation}\label{ex:g3c1monos}
f_3, f_4 a_1, a_1^2 f_5, a_1^3 f_6, a_1^4 f_7, f_5 a_2, a_1 a_2 f_6, a_1^2 
a_2 f_7, a_2^2 f_7, f_6 a_3, a_1 a_3 f_7, a_1 b_0^2, b_0 b_1\text{.}
\end{equation}

Next, we call the distillation algorithm using this list of monomials;
we can combine these two steps into one by using the following:
\[{\btt{}\maple{distillation(7,get\_monos(7,5,2));}}\]

The distillation process starts off by separating our list of monomials
into separate levels, so each expression is indexed by the order of
the pole at $\infty$ of that expression. The aim is then to find linear
combinations of expressions in any particular level that sit in a lower
level. Our list of monomials separates into levels as indicated by the
table in Figure~\ref{fig:g3exlevels}.

\begin{figure}[h]
 \begin{center}
  \begin{tabular}{|c|l|}
   \hline Level & Expressions\\\hline
    $8$ & $f_7a_1^4$,$a_1b_0^2$\\
    $6$ & $f_6a_1^3$, $f_7a_1^2a_2$, $b_0b_1$\\
    $4$ & $f_5a_1^2$, $f_6a_1a_2$, $f_7a_2^2$, $f_7a_1a_3$\\
    $2$ & $f_4a_1$, $f_5a_2$, $f_6a_3$\\
    $0$ & $f_3$\\\hline
  \end{tabular}
 \end{center}
 \caption{Monomials indexed by pole-order at $\infty$}\label{fig:g3exlevels}
\end{figure}

Having separated these expressions, the procedure now iterates through
the list, from the highest level (in this case, 8) to the lowest level
seeking linear combinations of the expressions from that level that
belong in a lower level. When these combinations are found, the
resulting expression is added to that (possibly new) lower level. Note
that, as any newly created level is always lower than the level
currently being processed, and that below 0 only the level $-\infty$
exists\footnote{The level $-\infty$
 represents those expressions which are identically 0.},
this algorithm always terminates.

Figure~\ref{fig:g3dist} indicates the progress of the distillation as
each level is processed. Each entry in the first `Level' column
represents the processing of a particular level, so that anything that
appears in the same row was as a result of processing that particular
level. The `Terms' column represents the total number of terms in all
expressions under consideration expanded into the corresponding
expression in $x_i$, $y_i$; e.g., $a_1$ expands into $x_1+x_2+x_3$ in
genus 3, and so has 3 terms, but $a_3$ expands into $x_1x_2x_3$ and
has only 1 term.  The `Matrix' column represents the dimension of the
matrix used to calculate the basis for the linear combinations of
expressions of a particular level that would sit in a lower
level. Finally, the `Added Expressions' columns represent the results
of processing the current level; an entry in the `Expressions' column
represents an expression that was obtained from the current level that
sits in a lower level, which is indicated in the adjacent `Level'
column.

\newcolumntype{d}[1]{D{*}{\times}{#1}}
\begin{figure}[h]
 \begin{center}
  \begin{tabular}{|c|c|d{-1}|c|l|}
   \hline & &\multicolumn{1}{c|}{} 
               &\multicolumn{2}{c|}{Added Expressions}\\\cline{4-5}
   Level & Terms & \multicolumn{1}{c|}{Matrix} & Level & Expressions\\
   \hline\hline
     8   &  205 &  3*2 & 6 & $a_1b_0^2-f_7a_1^4$\\\hline
     6   &  353 &  7*4 & 4 & $a_1b_0^2-f_7a_1^4+f_7a_1^2a_2-f_6a_1^3$\\\cline{4-5}
         &      &      & 4 & $b_0b_1+f_7a_1^2a_2$\\\hline
     4   &  363 & 12*6 & 3 & $-f_5 a_1^2-f_7 a_2^2+f_7 a_1 a_3-f_6 a_1^3$\\
         &      &      &   & $\quad+f_7 a_1^2 a_2-f_7 a_1^4+a_1 b_0^2$\\\cline{4-5}
         &      &      & 3 & $f_6 a_1 a_2-f_7 a_1 a_3+f_7 a_1^2 a_2+b_0 b_1$\\\hline
     3   &  264 &  4*2 & 2 & $-f_5 a_1^2-f_7 a_2^2-f_7 a_1 a_3-f_6 a_1^3$\\
         &      &      &   & $\quad+3 f_7 a_1^2 a_2-f_7 a_1^4+a_1 b_0^2$\\
         &      &      &   & $\quad+2 f_6 a_1 a_2+2 b_0 b_1$\\\hline
     2   &  201 & 14*4 &\multicolumn{2}{c|}{---}\\\hline
     0   &   19 &  3*1 &\multicolumn{2}{c|}{---}\\\hline
  \end{tabular}
 \end{center}
 \caption{Distillation to obtain $c_1$ in genus 3.}\label{fig:g3dist}
\end{figure}

So, for instance, whilst processing level 8, the expanded expressions
under consideration had 205 terms in total; a $3\times2$ matrix was used
to calculate that $a_1b_0^2-f_7a_1^4$ should sit in level $6$. Note
in particular that, before level 4 was processed, level 3 did not exist.
After level 4 was processed, the next-heighest level was the newly-created
level 3, and so that was the next level to be processed.

This process continues until finally we reach the state indicated by
Figure~\ref{fig:g3postlevels}; at this point, we can see that $c_1$ is
the new expression in level 2, and our example is complete.

\begin{figure}[h]
 \begin{center}
  \begin{tabular}{|c|l|}
   \hline Level & Expressions\\\hline
    $8$ & $f_7a_1^4$,$a_1b_0^2$\\\hline
    $6$ & $f_6a_1^3$, $f_7a_1^2a_2$, $b_0b_1$, $-f_7 a_1^4+a_1 b_0^2$\\\hline
    $4$ & $f_5a_1^2$, $f_6a_1a_2$, $f_7a_2^2$, $f_7a_1a_3$,\\
        & $-f_6 a_1^3+f_7 a_1^2 a_2-f_7 a_1^4+a_1 b_0^2$,\\
        & $f_7 a_1^2 a_2+b_0 b_1$\\\hline
    $3$ & $-f_5 a_1^2-f_7 a_2^2+f_7 a_1 a_3-f_6 a_1^3+f_7 a_1^2 a_2-f_7 a_1^4+a_1 b_0^2$,\\
        & $f_6 a_1 a_2-f_7 a_1 a_3+f_7 a_1^2 a_2+b_0 b_1$\\\hline
    $2$ & $-f_5 a_1^2-f_7 a_2^2-f_7 a_1 a_3-f_6 a_1^3+3 f_7 a_1^2 a_2$\\
        & $\quad -f_7 a_1^4+a_1 b_0^2+2 f_6 a_1 a_2+2 b_0 b_1$,\\
        & $f_4a_1$, $f_5a_2$, $f_6a_3$\\\hline
    $0$ & $f_3$\\\hline
  \end{tabular}
 \end{center}
 \caption{Post-distillation: expressions indexed by pole-order at $\infty$}
\label{fig:g3postlevels}
\end{figure}

It may be instructive to give some indication of how the size of
objects that occur using this algorithm increases as the task becomes
more complex. Contrast the output from Figure~\ref{fig:g3dist} with
that from Figure~\ref{fig:g3dist2} to see how, in calculating $c_3$
(which has the same $y$-weight and a $w$-weight greater by only 2),
there is a large increase in the size of objects needed. Note that, as
the expressions are very long, the `Expressions' column has been
replaced in Figure~\ref{fig:g3dist2} by a column indicating the
\emph{number} of expressions that are found to be in a lower level.

\begin{figure}[h]
 \begin{center}
  \begin{tabular}{|c|c|d{-1}|c|c|}
   \hline & &\multicolumn{1}{c|}{} 
               &\multicolumn{2}{c|}{Added Expressions}\\\cline{4-5}
   Level & Terms & \multicolumn{1}{c|}{Matrix} & Level & \#(Expressions)\\
   \hline\hline
    12 &  828 &  4*4  & 10 & 3\\\hline
    10 & 1682 &  7*8  &  8 & 6\\\hline
     8 & 2411 & 12*14 &  7 & 4\\\cline{4-5}
       &      &       &  6 & 6\\\hline
     7 & 1042 &  4*4  &  6 & 3\\\hline
     6 & 2178 & 18*16 &  5 & 8\\\cline{4-5}
       &      &       &  4 & 2\\\hline
     5 & 1366 &  6*8  &  4 & 4\\\cline{4-5}
       &      &       & $-\infty$ & 2\\\hline
     4 & 1140 & 21*12 &  3 & 4\\\hline
     3 &  481 &  4*4  &  2 & 3\\\hline
     2 &  397 & 19*6  &  0 & 1\\\cline{4-5}
       &      &       & $-\infty$ & 1\\\hline
     0 &   38 &  3*2  & $-\infty$ & 1\\\hline
  \end{tabular}
 \end{center}
 \caption{Distillation to obtain $c_3$ in genus 3.}\label{fig:g3dist2}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \input algodef.tex

\chapter{Algorithms for Determining Defining Equations}
\label{ch:algodef}
%Methods for finding defining equations (more programs).
%Give actual L(2*Theta) defining equations.

\section{Outline of General Strategy}\label{sec:defstrat}

To find the various relations, we simply seek to find linear combinations
of products of the $a_i, c_i$ in such a way as to clear the coefficient of
$y_iy_j$; then whatever remains may be
expressed in terms of the elementary symmetric polynomials in the $x_i$
which, happily, are easily expressed in terms of the $a_i$.

We first give a systematic way of deriving the quartic defining equation
of the Kummer surface in genus~2.
We can first look at the set of quartic monomials
in $\ka_0, \ldots , \ka_3$, as defined in~(\ref{eq:gen2kum}),
and draw up a chart of coefficients of $y_1 y_2$. We find linear
dependency, that
\begin{equation}\label{eq:lindep}
\begin{array}{rcl}
&\ka_1^2 \ka_3^2 - 4 \ka_0 \ka_2 \ka_3^2
- 4 f_0 \ka_0^3 \ka_3 - 2 f_1 \ka_0^2 \ka_1 \ka_3 
- 4 f_2 \ka_0^2 \ka_2 \ka_3\\ 
&- 2 f_3 \ka_0 \ka_1 \ka_2 \ka_3
- 4 f_4 \ka_0 \ka_2^2 \ka_3
- 2 f_5 \ka_1 \ka_2^2 \ka_3 - 4 f_6 \ka_2^3 \ka_3\\
\end{array}
\end{equation}
has coefficient of $y_1 y_2$ equal to~0. The resulting
rational function has denomiator a power of~$\delta$ and
numerator entirely in terms of~$x_1,x_2$. Since there
is a pole only at~$(x,u) = (x_1,-y_1)$ but not at
$(x,u) = (x_1,y_1)$, it follows that the denominator must
cancel, and we are left with a symmetric polynomial
in $x_1,x_2$, which can be expressed in terms of
the elementary symmetric polynomials $\ka_0,\ka_1,\ka_2 = a_0,a_1,a_2$. 
This then gives the quartic defining equation~(\ref{eq:kummdefn})
for the Kummer surface. Note that the defining equation is only
quadratic in~$\ka_3$, the unique function with denominator~$\delta$. 

In genus~3,
in order to clear the $y_iy_j$ coefficients, we first note that each
$y_iy_j$ coefficient is the same in $\gamma_i$ as the
corresponding coefficient in $c_i = \wred \gamma_i$
so it is just as easy (and may be preferable) to perform the initial
stage using the $\gamma_i$ rather than the $c_i$.
Define, as usual,
$\delta=(x_1-x_2)(x_2-x_3)(x_3-x_1)$ then the following collections
should be sufficient to determine all the required relations.

\begin{align}
T_1 &= \{\delta^4a_ia_ja_kc_\ell:3\geq i\geq j\geq k\geq 0;3\geq\ell\geq 0\}\\
T_2 &= \{\delta^4a_ia_jc_kc_\ell:3\geq i\geq j\geq 0;3\geq k\geq \ell\geq 0\}\\
\end{align}

Our general strategy is then to run through a chart of coefficients
of $y_1 y_2$ and to find linear dependencies between them. Note that,
by symmetry, if the coefficient of $y_1 y_2$ is forced to be~0,
then so will be the coefficients of $y_2 y_3$ and $y_3 y_1$, giving
a numerator entirely in terms of $x_1,x_2,x_3$. The resulting
symmetric polynomial in $x_1,x_2,x_3$ is then expressed in
terms of the elementary symmetric functions $a_0,a_1,a_2$.

\section{Defining Equations for $L(2\Theta)$ in Genus~$3$.}\label{sec:defL2T}

In genus~3, the expressions are too large, and the available
monomials are too numerous, to search for linear relationships
between the entire set of (for example) quadratic, cubic and
quartic monomials. It is helpful to
have a technical lemma, analogous to Lemma~\ref{lem:wtslnt}.

\begin{lemma}\label{lem:wtsdef}
Let $\hbox{wt}$ be a weight for which $\C : y^2 = f_n x^n + \ldots + f_0$,
$\delta$ and each member $d_1,\ldots d_k$ of a basis
for $L(n\Theta)$ are homogeneous (where the denominator
of each $d_i$ is a power of~$\delta$).
Let $\phi(d_1,\ldots ,d_k) = 0$ be a polynomial relation
satisfied by $d_1,\ldots ,d_k$.
Let $\phi_{i_1}, \ldots ,\phi_{i_m}$
be the homogeneous pieces of~$\phi$ satisfying $\phi = \phi_{i_1}+ \ldots +\phi_{i_m}$
and $\hbox{wt}(\phi_{i_1}) = i_1, \ldots , \hbox{wt}(\phi_{i_m}) = i_m$,
where $i_1,\ldots ,i_m \in \bbbz$ are distinct.
Then each of $\phi_{i_1}, \ldots ,\phi_{i_m}$ is also
a relation satisfied by $d_1,\ldots d_k$.
\end{lemma}

\begin{proof}
\ The same argument applies as in the proof of
Lemma~\ref{lem:wtslnt}, since
polynomial relations satisfied by $d_1,\ldots d_k$ 
are preserved by linear combinations.
\end{proof}

This means that, when searching for a polynomial
relation of degree~$\ell$, it is not necessary to search
for linear dependencies on the set of all monomials of
degree~$\ell$ in $d_1,\ldots d_k$; rather, one can slice
up the set of such monomials in portions of equal weight,
for any weight satisfying the conditions of Lemma~\ref{lem:wtsdef},
such that the $x$-weight and $y$-weight of Figure~\ref{fig:wam}.

After considerable computations, we can exhaustively
find every possible quadratic, cubic and quartic relation
on the Kummer, and so list a set of
defining equations for the $L(2\Theta)$ embedding of
the Kummer variety in genus~3, subject to Conjecture~\ref{conj:kum}.
Redundant equations can be weeded out using the Groebner
basis package in Maple. If $r_k$ lies in the variety
defined by $r_1,\ldots ,r_{k-1}$ then the Groebner basis
of $r_1,\ldots ,r_{k-1}$ is the same as that
of $r_1,\ldots ,r_{k-1},r_k$, and so we can sometimes notice
and remove any redundant relations, which we have
done from an original list of~41. However, the
equations are too large 
to perform a Groebner basis operation on the whole set
of polynomials; we conjecture that there are no dependencies
based on interactive sessions in Maple applied
to many specialisations of~$f_0,\ldots ,f_7$.
The final result gives
the genus~3 equivalent of the single quartic
equation~(\ref{eq:kummdefn}) whch defines the
Kummer surface in genus~2.

\begin{theorem}\label{thm:kummdefg3}
Let $a_0,a_1,a_2,a_3,c_0,c_1,c_2,c_3$ be the basis
for $L(2\Theta)$ in genus~3 given in Theorem~\ref{thm:g3kum}.
Then the following ${\bf eq[i]} = 0$
generate all relations that are
at most quartic in $a_0,a_1,a_2,a_3$, $c_0,c_1,c_2,c_3$,
and are at most quadratic in $c_0,c_1,c_2,c_3$.
\end{theorem}

{\scriptsize
\raggedright
\sloppy
\noindent
${\bf eq[1]} =
a_{{0}}c_{{3}}$ $-a_{{1}}c_{{2}}$ $-a_{{2}}c_{{1}}$ $-a_{{3}}c_{{0}}
-2\,f_{{5}}a_{{1}}a_{{3}}$ $+f_{{5}}{a_{{2}}}^{2}$ $+2\,f_{{6}}a_{{2}}a_{{3}}$ $+3\,f_{{7}}{a_{{
3}}}^{2}$.

\noindent
${\bf eq[2]} =
-4\,a_{{1}}a_{{3}}{c_{{3}}}^{2}$ $+{a_{{2}}}^{2}{c_{{3}}}^{2}$ $+2\,a_{{2}}a_{{3}}c_{{2}}c_{{
3}}$ $+4\,{a_{{3}}}^{2}c_{{1}}c_{{3}}$ $+{a_{{3}}}^{2}{c_{{2}}}^{2}
-4\,f_{{0}}{a_{{0}}}^{2}a_{{3}}c_{{3}}$ $+4\,f_{{0}}a_{{0}}a_{{1}}a_{{2}}c_{{3}}$ $+4\,f_{{0}
}a_{{0}}a_{{1}}a_{{3}}c_{{2}}$ $-4\,f_{{0}}a_{{0}}{a_{{2}}}^{2}c_{{2}}$ $-4\,f_{{0}}{a_{{1}}}
^{3}c_{{3}}$ $+4\,f_{{0}}{a_{{1}}}^{2}a_{{2}}c_{{2}}$ $+4\,f_{{0}}{a_{{1}}}^{2}a_{{3}}c_{{1}}
+8\,f_{{0}}a_{{1}}a_{{2}}a_{{3}}c_{{0}}$ $-4\,f_{{0}}{a_{{2}}}^{3}c_{{0}}$ $+2\,f_{{1}}a_{{0}
}{a_{{2}}}^{2}c_{{3}}$ $-2\,f_{{1}}a_{{0}}a_{{2}}a_{{3}}c_{{2}}$ $-2\,f_{{1}}{a_{{1}}}^{2}a_{
{2}}c_{{3}}$ $+2\,f_{{1}}{a_{{1}}}^{2}a_{{3}}c_{{2}}$ $+4\,f_{{1}}a_{{1}}{a_{{3}}}^{2}c_{{0}}
-4\,f_{{1}}{a_{{2}}}^{2}a_{{3}}c_{{0}}$ $-2\,f_{{2}}a_{{0}}{a_{{3}}}^{2}c_{{2}}$ $+4\,f_{{2}}
{a_{{1}}}^{2}a_{{3}}c_{{3}}$ $-2\,f_{{2}}a_{{1}}{a_{{2}}}^{2}c_{{3}}$ $-4\,f_{{2}}a_{{1}}{a_{
{3}}}^{2}c_{{1}}$ $+2\,f_{{2}}{a_{{2}}}^{2}a_{{3}}c_{{1}}$ $-2\,f_{{2}}a_{{2}}{a_{{3}}}^{2}c_
{{0}}$ $-4\,f_{{3}}a_{{0}}{a_{{3}}}^{2}c_{{3}}$ $+6\,f_{{3}}a_{{1}}a_{{2}}a_{{3}}c_{{3}}$ $+2\,f
_{{3}}a_{{1}}{a_{{3}}}^{2}c_{{2}}$ $-2\,f_{{3}}{a_{{2}}}^{3}c_{{3}}$ $-2\,f_{{3}}{a_{{2}}}^{2
}a_{{3}}c_{{2}}$ $+4\,f_{{5}}{a_{{3}}}^{3}c_{{2}}$ $-4\,f_{{6}}{a_{{3}}}^{3}c_{{3}}
+
8\,f_{{0}}f_{{5}}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $-8\,f_{{0}}f_{{5}}a_{{0}}{a_{{2}}}^{2}a_{{3
}}$ $+4\,f_{{0}}f_{{5}}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $-4\,f_{{0}}f_{{6}}{a_{{1}}}^{2}{a_{{3}}}
^{2}$ $-8\,f_{{0}}f_{{6}}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $+4\,f_{{0}}f_{{6}}{a_{{2}}}^{4}$ $-24\,f_
{{0}}f_{{7}}a_{{1}}a_{{2}}{a_{{3}}}^{2}$ $+12\,f_{{0}}f_{{7}}{a_{{2}}}^{3}a_{{3}}$ $+{f_{{1}}
}^{2}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $-2\,{f_{{1}}}^{2}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $+{f_{{1}}}^{
2}{a_{{1}}}^{4}$ $+2\,f_{{1}}f_{{2}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $-2\,f_{{1}}f_{{2}}a_{{0}}{a
_{{1}}}^{2}a_{{3}}$ $-2\,f_{{1}}f_{{2}}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $+2\,f_{{1}}f_{{2}}{a_{{1
}}}^{3}a_{{2}}$ $+2\,f_{{1}}f_{{3}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $-2\,f_{{1}}f_{{3}}a_{{0}}{a
_{{2}}}^{3}$ $-2\,f_{{1}}f_{{3}}{a_{{1}}}^{3}a_{{3}}$ $+2\,f_{{1}}f_{{3}}{a_{{1}}}^{2}{a_{{2}
}}^{2}$ $-4\,f_{{1}}f_{{5}}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $+4\,f_{{1}}f_{{5}}{a_{{1}}}^{2}{a_{{
3}}}^{2}$ $-4\,f_{{1}}f_{{6}}a_{{1}}a_{{2}}{a_{{3}}}^{2}$ $+4\,f_{{1}}f_{{6}}{a_{{2}}}^{3}a_{
{3}}$ $-12\,f_{{1}}f_{{7}}a_{{1}}{a_{{3}}}^{3}$ $+12\,f_{{1}}f_{{7}}{a_{{2}}}^{2}{a_{{3}}}^{2
}$ $+{f_{{2}}}^{2}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $-2\,{f_{{2}}}^{2}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $+
{f_{{2}}}^{2}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $+2\,f_{{2}}f_{{3}}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $-2
\,f_{{2}}f_{{3}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $-2\,f_{{2}}f_{{3}}{a_{{1}}}^{2}a_{{2}}a_{{3}
}$ $+2\,f_{{2}}f_{{3}}a_{{1}}{a_{{2}}}^{3}$ $-4\,f_{{2}}f_{{5}}a_{{0}}{a_{{3}}}^{3}$ $+4\,f_{{2}
}f_{{5}}a_{{1}}a_{{2}}{a_{{3}}}^{2}$ $-2\,f_{{2}}f_{{5}}{a_{{2}}}^{3}a_{{3}}$ $+4\,f_{{2}}f_{
{6}}a_{{1}}{a_{{3}}}^{3}$ $+6\,f_{{2}}f_{{7}}a_{{2}}{a_{{3}}}^{3}$ $+{f_{{3}}}^{2}{a_{{1}}}^{
2}{a_{{3}}}^{2}$ $-2\,{f_{{3}}}^{2}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $+{f_{{3}}}^{2}{a_{{2}}}^{4}$ $+
4\,f_{{3}}f_{{5}}a_{{1}}{a_{{3}}}^{3}$ $-4\,f_{{3}}f_{{5}}{a_{{2}}}^{2}{a_{{3}}}^{2}$ $+4\,{f
_{{5}}}^{2}{a_{{3}}}^{4}$.

\noindent
$\bf{eq[3]} =
-a_{{0}}a_{{3}}{c_{{3}}}^{2}$ $-3\,a_{{1}}a_{{3}}c_{{2}}c_{{3}}$ $+{a_{{2}}}^{2}c_{{2}}c_{{3}
}$ $+a_{{2}}a_{{3}}{c_{{2}}}^{2}$ $-{a_{{3}}}^{2}c_{{0}}c_{{3}}$ $+{a_{{3}}}^{2}c_{{1}}c_{{2}}
+
4\,f_{{0}}{a_{{0}}}^{2}a_{{2}}c_{{3}}$ $-4\,f_{{0}}a_{{0}}{a_{{1}}}^{2}c_{{3}}$ $+2\,f_{{0}}a
_{{0}}a_{{1}}a_{{3}}c_{{1}}$ $-2\,f_{{0}}a_{{0}}{a_{{2}}}^{2}c_{{1}}$ $+2\,f_{{0}}{a_{{1}}}^{
2}a_{{3}}c_{{0}}$ $-2\,f_{{0}}a_{{1}}{a_{{2}}}^{2}c_{{0}}$ $+f_{{1}}{a_{{0}}}^{2}a_{{3}}c_{{3
}}$ $-f_{{1}}a_{{0}}a_{{1}}a_{{2}}c_{{3}}$ $+f_{{1}}a_{{0}}{a_{{2}}}^{2}c_{{2}}$ $+f_{{1}}a_{{0}
}{a_{{3}}}^{2}c_{{0}}$ $-f_{{1}}{a_{{1}}}^{2}a_{{2}}c_{{2}}$ $-f_{{1}}{a_{{1}}}^{2}a_{{3}}c_{
{1}}$ $-2\,f_{{1}}a_{{1}}a_{{2}}a_{{3}}c_{{0}}$ $-2\,f_{{2}}a_{{0}}{a_{{2}}}^{2}c_{{3}}$ $-f_{{2
}}a_{{0}}{a_{{3}}}^{2}c_{{1}}$ $+4\,f_{{2}}{a_{{1}}}^{2}a_{{3}}c_{{2}}$ $+3\,f_{{2}}a_{{1}}a_
{{2}}a_{{3}}c_{{1}}$ $+2\,f_{{2}}a_{{1}}{a_{{3}}}^{2}c_{{0}}$ $+f_{{2}}{a_{{2}}}^{3}c_{{1}}$ $+f
_{{2}}{a_{{2}}}^{2}a_{{3}}c_{{0}}$ $+3\,f_{{3}}a_{{0}}a_{{2}}a_{{3}}c_{{3}}$ $+f_{{3}}a_{{1}}
{a_{{3}}}^{2}c_{{1}}$ $-f_{{3}}{a_{{2}}}^{3}c_{{2}}$ $-2\,f_{{3}}{a_{{2}}}^{2}a_{{3}}c_{{1}}$ $-
f_{{3}}a_{{2}}{a_{{3}}}^{2}c_{{0}}$ $+2\,f_{{4}}a_{{0}}{a_{{3}}}^{2}c_{{3}}$ $-3\,f_{{4}}a_{{
1}}a_{{2}}a_{{3}}c_{{3}}$ $-f_{{4}}a_{{1}}{a_{{3}}}^{2}c_{{2}}$ $+f_{{4}}{a_{{2}}}^{3}c_{{3}}
+f_{{4}}{a_{{2}}}^{2}a_{{3}}c_{{2}}$ $-8\,f_{{5}}a_{{1}}{a_{{3}}}^{2}c_{{3}}$ $+3\,f_{{5}}{a_
{{2}}}^{2}a_{{3}}c_{{3}}$ $+4\,f_{{5}}a_{{2}}{a_{{3}}}^{2}c_{{2}}$ $+2\,f_{{5}}{a_{{3}}}^{3}c
_{{1}}$ $+2\,f_{{6}}a_{{2}}{a_{{3}}}^{2}c_{{3}}$ $+5\,f_{{7}}{a_{{3}}}^{3}c_{{3}}
-2\,f_{{0}}f_{{5}}{a_{{1}}}^{3}a_{{3}}$ $+2\,f_{{0}}f_{{5}}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $-4\,f
_{{0}}f_{{6}}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $+4\,f_{{0}}f_{{6}}a_{{1}}{a_{{2}}}^{3}$ $-4\,f_{{0
}}f_{{7}}{a_{{1}}}^{2}{a_{{3}}}^{2}$ $+2\,f_{{0}}f_{{7}}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $+2\,f_{
{0}}f_{{7}}{a_{{2}}}^{4}$ $-{f_{{1}}}^{2}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $+{f_{{1}}}^{2}a_{{0}}{
a_{{1}}}^{3}$ $-f_{{1}}f_{{2}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $-f_{{1}}f_{{2}}{a_{{0}}}^{2}{a_{{
2}}}^{2}$ $+2\,f_{{1}}f_{{2}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $-f_{{1}}f_{{3}}a_{{0}}{a_{{1}}}^{2
}a_{{3}}$ $+f_{{1}}f_{{3}}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $-f_{{1}}f_{{4}}a_{{0}}a_{{1}}a_{{2}}a
_{{3}}$ $+f_{{1}}f_{{4}}a_{{0}}{a_{{2}}}^{3}$ $+f_{{1}}f_{{4}}{a_{{1}}}^{3}a_{{3}}$ $-f_{{1}}f_{
{4}}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $+f_{{1}}f_{{5}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $-2\,f_{{1}}f_{{
6}}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $+4\,f_{{1}}f_{{6}}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $-3\,f_{{1}}f
_{{7}}a_{{0}}{a_{{3}}}^{3}$ $+4\,f_{{1}}f_{{7}}a_{{1}}a_{{2}}{a_{{3}}}^{2}$ $+2\,f_{{1}}f_{{7
}}{a_{{2}}}^{3}a_{{3}}$ $-{f_{{2}}}^{2}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $+{f_{{2}}}^{2}a_{{0}}a_{
{1}}{a_{{2}}}^{2}$ $-f_{{2}}f_{{3}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $+f_{{2}}f_{{3}}a_{{0}}{a_{{
2}}}^{3}$ $-f_{{2}}f_{{4}}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $+f_{{2}}f_{{4}}a_{{0}}{a_{{2}}}^{2}a_
{{3}}$ $+f_{{2}}f_{{4}}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $-f_{{2}}f_{{4}}a_{{1}}{a_{{2}}}^{3}$ $+10\,
f_{{2}}f_{{5}}{a_{{1}}}^{2}{a_{{3}}}^{2}$ $-4\,f_{{2}}f_{{5}}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $-f
_{{2}}f_{{5}}{a_{{2}}}^{4}$ $-4\,f_{{2}}f_{{6}}a_{{1}}a_{{2}}{a_{{3}}}^{2}$ $-2\,f_{{2}}f_{{6
}}{a_{{2}}}^{3}a_{{3}}$ $-8\,f_{{2}}f_{{7}}a_{{1}}{a_{{3}}}^{3}$ $-f_{{2}}f_{{7}}{a_{{2}}}^{2
}{a_{{3}}}^{2}$ $-f_{{3}}f_{{4}}{a_{{1}}}^{2}{a_{{3}}}^{2}$ $+2\,f_{{3}}f_{{4}}a_{{1}}{a_{{2}
}}^{2}a_{{3}}$ $-f_{{3}}f_{{4}}{a_{{2}}}^{4}$ $-f_{{3}}f_{{5}}{a_{{2}}}^{3}a_{{3}}$ $+2\,f_{{3}}
f_{{6}}{a_{{2}}}^{2}{a_{{3}}}^{2}$ $+3\,f_{{3}}f_{{7}}a_{{2}}{a_{{3}}}^{3}$ $-2\,f_{{4}}f_{{5
}}a_{{1}}{a_{{3}}}^{3}$ $+2\,f_{{4}}f_{{5}}{a_{{2}}}^{2}{a_{{3}}}^{2}$ $+4\,{f_{{5}}}^{2}a_{{
2}}{a_{{3}}}^{3}$.

\noindent
${\bf eq[4]} =
-4\,a_{{0}}a_{{3}}c_{{2}}c_{{3}}$ $+{a_{{2}}}^{2}{c_{{2}}}^{2}$ $+2\,a_{{2}}a_{{3}}c_{{1}}c_{
{2}}$ $+{a_{{3}}}^{2}{c_{{1}}}^{2}
-4\,f_{{0}}{a_{{0}}}^{2}a_{{1}}c_{{3}}$ $+4\,f_{{0}}a_{{0}}a_{{1}}a_{{3}}c_{{0}}$ $-4\,f_{{0}
}a_{{0}}{a_{{2}}}^{2}c_{{0}}$ $-4\,f_{{1}}{a_{{0}}}^{2}a_{{2}}c_{{3}}$ $+2\,f_{{1}}a_{{0}}a_{
{1}}a_{{2}}c_{{2}}$ $-2\,f_{{1}}a_{{0}}a_{{1}}a_{{3}}c_{{1}}$ $+4\,f_{{1}}a_{{0}}{a_{{2}}}^{2
}c_{{1}}$ $+2\,f_{{2}}{a_{{0}}}^{2}a_{{3}}c_{{3}}$ $+2\,f_{{2}}a_{{0}}a_{{1}}a_{{3}}c_{{2}}$ $-2
\,f_{{2}}a_{{0}}{a_{{2}}}^{2}c_{{2}}$ $-2\,f_{{2}}a_{{0}}{a_{{3}}}^{2}c_{{0}}$ $-6\,f_{{4}}a_
{{0}}a_{{2}}a_{{3}}c_{{3}}$ $-2\,f_{{4}}a_{{1}}{a_{{3}}}^{2}c_{{1}}$ $+2\,f_{{4}}{a_{{2}}}^{3
}c_{{2}}$ $+4\,f_{{4}}{a_{{2}}}^{2}a_{{3}}c_{{1}}$ $+2\,f_{{4}}a_{{2}}{a_{{3}}}^{2}c_{{0}}$ $-12
\,f_{{5}}a_{{0}}{a_{{3}}}^{2}c_{{3}}$ $+4\,f_{{5}}{a_{{2}}}^{2}a_{{3}}c_{{2}}$ $+4\,f_{{5}}a_
{{2}}{a_{{3}}}^{2}c_{{1}}
-4\,f_{{0}}f_{{5}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $+4\,f_{{0}}f_{{5}}a_{{0}}a_{{1}}{a_{{2}}}^
{2}$ $-8\,f_{{0}}f_{{6}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $+8\,f_{{0}}f_{{6}}a_{{0}}{a_{{2}}}^{3}
-12\,f_{{0}}f_{{7}}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $+12\,f_{{0}}f_{{7}}a_{{0}}{a_{{2}}}^{2}a_
{{3}}$ $+{f_{{1}}}^{2}{a_{{0}}}^{2}{a_{{1}}}^{2}$ $+2\,f_{{1}}f_{{2}}{a_{{0}}}^{2}a_{{1}}a_{{
2}}$ $+2\,f_{{1}}f_{{4}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $-2\,f_{{1}}f_{{4}}a_{{0}}a_{{1}}{a_{{2}
}}^{2}$ $+8\,f_{{1}}f_{{5}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $-4\,f_{{1}}f_{{5}}a_{{0}}{a_{{2}}}^
{3}$ $+{f_{{2}}}^{2}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $+2\,f_{{2}}f_{{4}}a_{{0}}a_{{1}}a_{{2}}a_{{3
}}$ $-2\,f_{{2}}f_{{4}}a_{{0}}{a_{{2}}}^{3}$ $+8\,f_{{2}}f_{{5}}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $-6
\,f_{{2}}f_{{5}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $+4\,f_{{2}}f_{{6}}a_{{0}}a_{{2}}{a_{{3}}}^{2
}$ $+6\,f_{{2}}f_{{7}}a_{{0}}{a_{{3}}}^{3}$ $+{f_{{4}}}^{2}{a_{{1}}}^{2}{a_{{3}}}^{2}$ $-2\,{f_{
{4}}}^{2}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $+{f_{{4}}}^{2}{a_{{2}}}^{4}$ $+2\,f_{{4}}f_{{5}}{a_{{2
}}}^{3}a_{{3}}$ $-4\,f_{{4}}f_{{6}}{a_{{2}}}^{2}{a_{{3}}}^{2}$ $-6\,f_{{4}}f_{{7}}a_{{2}}{a_{
{3}}}^{3}$ $+4\,{f_{{5}}}^{2}{a_{{2}}}^{2}{a_{{3}}}^{2}$.

\noindent
${\bf eq[5]} =
-2\,a_{{0}}a_{{2}}{c_{{3}}}^{2}$ $-8\,a_{{0}}a_{{3}}c_{{2}}c_{{3}}$ $+2\,a_{{1}}a_{{2}}c_{{2}
}c_{{3}}$ $+4\,a_{{1}}a_{{3}}c_{{1}}c_{{3}}$ $+{a_{{2}}}^{2}{c_{{2}}}^{2}$ $+2\,{a_{{3}}}^{2}c_{
{0}}c_{{2}}$ $-3\,{a_{{3}}}^{2}{c_{{1}}}^{2}
-8\,f_{{0}}{a_{{0}}}^{2}a_{{1}}c_{{3}}$ $+8\,f_{{0}}{a_{{0}}}^{2}a_{{2}}c_{{2}}$ $+4\,f_{{0}}
{a_{{0}}}^{2}a_{{3}}c_{{1}}$ $-4\,f_{{0}}a_{{0}}{a_{{1}}}^{2}c_{{2}}$ $+4\,f_{{0}}a_{{0}}a_{{
1}}a_{{3}}c_{{0}}$ $+4\,f_{{0}}a_{{0}}{a_{{2}}}^{2}c_{{0}}$ $-4\,f_{{0}}{a_{{1}}}^{2}a_{{2}}c
_{{0}}$ $+2\,f_{{1}}{a_{{0}}}^{2}a_{{3}}c_{{2}}$ $-4\,f_{{1}}a_{{0}}a_{{1}}a_{{2}}c_{{2}}$ $-2\,
f_{{1}}a_{{0}}a_{{1}}a_{{3}}c_{{1}}$ $-4\,f_{{1}}a_{{0}}{a_{{2}}}^{2}c_{{1}}$ $+2\,f_{{1}}{a_
{{1}}}^{2}a_{{2}}c_{{1}}$ $-2\,f_{{1}}{a_{{1}}}^{2}a_{{3}}c_{{0}}$ $+2\,f_{{2}}a_{{0}}a_{{1}}
a_{{2}}c_{{3}}$ $+8\,f_{{2}}a_{{0}}a_{{1}}a_{{3}}c_{{2}}$ $-2\,f_{{2}}a_{{0}}{a_{{2}}}^{2}c_{
{2}}$ $+2\,f_{{2}}a_{{0}}{a_{{3}}}^{2}c_{{0}}$ $-2\,f_{{2}}{a_{{1}}}^{2}a_{{2}}c_{{2}}$ $+2\,f_{
{3}}a_{{0}}{a_{{2}}}^{2}c_{{3}}$ $+6\,f_{{3}}a_{{0}}a_{{2}}a_{{3}}c_{{2}}$ $+4\,f_{{3}}a_{{0}
}{a_{{3}}}^{2}c_{{1}}$ $-4\,f_{{3}}{a_{{1}}}^{2}a_{{3}}c_{{2}}$ $-6\,f_{{3}}a_{{1}}a_{{2}}a_{
{3}}c_{{1}}$ $-2\,f_{{3}}a_{{1}}{a_{{3}}}^{2}c_{{0}}$ $+2\,f_{{3}}{a_{{2}}}^{3}c_{{1}}$ $+2\,f_{
{3}}{a_{{2}}}^{2}a_{{3}}c_{{0}}$ $-6\,f_{{4}}a_{{0}}a_{{2}}a_{{3}}c_{{3}}$ $+2\,f_{{4}}a_{{0}
}{a_{{3}}}^{2}c_{{2}}$ $-4\,f_{{4}}{a_{{1}}}^{2}a_{{3}}c_{{3}}$ $+2\,f_{{4}}a_{{1}}{a_{{2}}}^
{2}c_{{3}}$ $+2\,f_{{4}}a_{{1}}{a_{{3}}}^{2}c_{{1}}$ $+2\,f_{{4}}{a_{{2}}}^{3}c_{{2}}$ $+2\,f_{{
4}}{a_{{2}}}^{2}a_{{3}}c_{{1}}$ $+4\,f_{{4}}a_{{2}}{a_{{3}}}^{2}c_{{0}}$ $-16\,f_{{5}}a_{{0}}
{a_{{3}}}^{2}c_{{3}}$ $+2\,f_{{5}}a_{{1}}a_{{2}}a_{{3}}c_{{3}}$ $-6\,f_{{5}}a_{{1}}{a_{{3}}}^
{2}c_{{2}}$ $+6\,f_{{5}}{a_{{2}}}^{2}a_{{3}}c_{{2}}$ $+4\,f_{{6}}{a_{{3}}}^{3}c_{{1}}$ $+2\,f_{{
7}}a_{{2}}{a_{{3}}}^{2}c_{{3}}$ $-4\,f_{{7}}{a_{{3}}}^{3}c_{{2}}
+
16\,f_{{0}}f_{{5}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $-16\,f_{{0}}f_{{5}}a_{{0}}{a_{{1}}}^{2}a_{
{3}}$ $+4\,f_{{0}}f_{{5}}{a_{{1}}}^{3}a_{{2}}$ $-16\,f_{{0}}f_{{6}}a_{{0}}a_{{1}}a_{{2}}a_{{3
}}$ $+8\,f_{{0}}f_{{6}}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $-8\,f_{{0}}f_{{7}}a_{{0}}a_{{1}}{a_{{3}}}
^{2}$ $-16\,f_{{0}}f_{{7}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $+8\,f_{{0}}f_{{7}}{a_{{1}}}^{2}a_{{2}
}a_{{3}}$ $+4\,f_{{0}}f_{{7}}a_{{1}}{a_{{2}}}^{3}$ $-2\,{f_{{1}}}^{2}{a_{{0}}}^{3}a_{{2}}$ $+3\,
{f_{{1}}}^{2}{a_{{0}}}^{2}{a_{{1}}}^{2}$ $-2\,f_{{1}}f_{{2}}{a_{{0}}}^{3}a_{{3}}$ $+4\,f_{{1}
}f_{{2}}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $-2\,f_{{1}}f_{{3}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $+4\,f_{
{1}}f_{{3}}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $-2\,f_{{1}}f_{{3}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $-2\,f
_{{1}}f_{{4}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $+4\,f_{{1}}f_{{4}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $-2
\,f_{{1}}f_{{4}}{a_{{1}}}^{3}a_{{2}}$ $+4\,f_{{1}}f_{{5}}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $-6\,f_{
{1}}f_{{5}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $+4\,f_{{1}}f_{{5}}a_{{0}}{a_{{2}}}^{3}$ $+2\,f_{{1}
}f_{{5}}{a_{{1}}}^{3}a_{{3}}$ $-2\,f_{{1}}f_{{5}}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $-4\,f_{{1}}f_{{
6}}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $+8\,f_{{1}}f_{{6}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $+4\,f_{{1}}f
_{{6}}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $-2\,f_{{1}}f_{{7}}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $+4\,f_{{1
}}f_{{7}}{a_{{1}}}^{2}{a_{{3}}}^{2}$ $+4\,f_{{1}}f_{{7}}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $+{f_{{2
}}}^{2}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $+2\,f_{{2}}f_{{3}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $-2\,f_{{2
}}f_{{3}}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $-2\,f_{{2}}f_{{4}}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $+6\,f_{
{2}}f_{{4}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $-2\,f_{{2}}f_{{4}}a_{{0}}{a_{{2}}}^{3}$ $-2\,f_{{2}
}f_{{4}}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $+22\,f_{{2}}f_{{5}}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $-6\,f_{
{2}}f_{{5}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $-6\,f_{{2}}f_{{5}}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $-8\,
f_{{2}}f_{{7}}a_{{0}}{a_{{3}}}^{3}$ $+2\,f_{{2}}f_{{7}}a_{{1}}a_{{2}}{a_{{3}}}^{2}$ $+2\,{f_{
{3}}}^{2}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $-2\,{f_{{3}}}^{2}a_{{0}}{a_{{2}}}^{3}$ $-2\,f_{{3}}f_
{{4}}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $+2\,f_{{3}}f_{{4}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $+2\,f_{{3}
}f_{{4}}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $-2\,f_{{3}}f_{{4}}a_{{1}}{a_{{2}}}^{3}$ $+12\,f_{{3}}f_
{{5}}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $-10\,f_{{3}}f_{{5}}{a_{{1}}}^{2}{a_{{3}}}^{2}$ $+4\,f_{{3}
}f_{{5}}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $-2\,f_{{3}}f_{{5}}{a_{{2}}}^{4}$ $+4\,f_{{3}}f_{{6}}a_{
{1}}a_{{2}}{a_{{3}}}^{2}$ $-4\,f_{{3}}f_{{6}}{a_{{2}}}^{3}a_{{3}}$ $+8\,f_{{3}}f_{{7}}a_{{1}}
{a_{{3}}}^{3}$ $-8\,f_{{3}}f_{{7}}{a_{{2}}}^{2}{a_{{3}}}^{2}$ $+{f_{{4}}}^{2}{a_{{1}}}^{2}{a_
{{3}}}^{2}$ $-2\,{f_{{4}}}^{2}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $+{f_{{4}}}^{2}{a_{{2}}}^{4}$ $+4\,f_
{{4}}f_{{5}}a_{{0}}{a_{{3}}}^{3}$ $-4\,f_{{4}}f_{{5}}a_{{1}}a_{{2}}{a_{{3}}}^{2}$ $+4\,f_{{4}
}f_{{5}}{a_{{2}}}^{3}a_{{3}}$ $-4\,f_{{4}}f_{{6}}a_{{1}}{a_{{3}}}^{3}$ $-4\,f_{{4}}f_{{6}}{a_
{{2}}}^{2}{a_{{3}}}^{2}$ $-12\,f_{{4}}f_{{7}}a_{{2}}{a_{{3}}}^{3}$ $-12\,{f_{{5}}}^{2}a_{{1}}
{a_{{3}}}^{3}$ $+12\,{f_{{5}}}^{2}{a_{{2}}}^{2}{a_{{3}}}^{2}$ $+8\,f_{{5}}f_{{6}}a_{{2}}{a_{{
3}}}^{3}$ $+4\,f_{{5}}f_{{7}}{a_{{3}}}^{4}$.

\noindent
${\bf eq[6]}=
2\,a_{{0}}a_{{3}}c_{{2}}c_{{3}}$ $+2\,a_{{1}}a_{{3}}{c_{{2}}}^{2}$ $-{a_{{2}}}^{2}{c_{{2}}}^{
2}$ $+2\,{a_{{3}}}^{2}c_{{0}}c_{{2}}$ $-{a_{{3}}}^{2}{c_{{1}}}^{2}
+
4\,f_{{0}}{a_{{0}}}^{2}a_{{1}}c_{{3}}$ $-4\,f_{{0}}a_{{0}}a_{{1}}a_{{3}}c_{{0}}$ $+4\,f_{{0}}
a_{{0}}{a_{{2}}}^{2}c_{{0}}$ $+4\,f_{{1}}{a_{{0}}}^{2}a_{{2}}c_{{3}}$ $-2\,f_{{1}}a_{{0}}a_{{
1}}a_{{2}}c_{{2}}$ $+2\,f_{{1}}a_{{0}}a_{{1}}a_{{3}}c_{{1}}$ $-4\,f_{{1}}a_{{0}}{a_{{2}}}^{2}
c_{{1}}$ $-2\,f_{{2}}{a_{{0}}}^{2}a_{{3}}c_{{3}}$ $-2\,f_{{2}}a_{{0}}a_{{1}}a_{{3}}c_{{2}}$ $+2
\,f_{{2}}a_{{0}}{a_{{2}}}^{2}c_{{2}}$ $+2\,f_{{2}}a_{{0}}{a_{{3}}}^{2}c_{{0}}$ $+6\,f_{{4}}a_
{{0}}a_{{2}}a_{{3}}c_{{3}}$ $+2\,f_{{4}}a_{{1}}{a_{{3}}}^{2}c_{{1}}$ $-2\,f_{{4}}{a_{{2}}}^{3
}c_{{2}}$ $-4\,f_{{4}}{a_{{2}}}^{2}a_{{3}}c_{{1}}$ $-2\,f_{{4}}a_{{2}}{a_{{3}}}^{2}c_{{0}}$ $+12
\,f_{{5}}a_{{0}}{a_{{3}}}^{2}c_{{3}}$ $+4\,f_{{5}}a_{{1}}{a_{{3}}}^{2}c_{{2}}$ $-6\,f_{{5}}{a
_{{2}}}^{2}a_{{3}}c_{{2}}$ $-4\,f_{{5}}a_{{2}}{a_{{3}}}^{2}c_{{1}}$ $-4\,f_{{6}}a_{{2}}{a_{{3
}}}^{2}c_{{2}}$ $-6\,f_{{7}}{a_{{3}}}^{3}c_{{2}}
+
4\,f_{{0}}f_{{5}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $-4\,f_{{0}}f_{{5}}a_{{0}}a_{{1}}{a_{{2}}}^{
2}$ $+8\,f_{{0}}f_{{6}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $-8\,f_{{0}}f_{{6}}a_{{0}}{a_{{2}}}^{3}$ $+
12\,f_{{0}}f_{{7}}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $-12\,f_{{0}}f_{{7}}a_{{0}}{a_{{2}}}^{2}a_{
{3}}$ $-{f_{{1}}}^{2}{a_{{0}}}^{2}{a_{{1}}}^{2}$ $-2\,f_{{1}}f_{{2}}{a_{{0}}}^{2}a_{{1}}a_{{2
}}$ $-2\,f_{{1}}f_{{4}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $+2\,f_{{1}}f_{{4}}a_{{0}}a_{{1}}{a_{{2}}
}^{2}$ $-8\,f_{{1}}f_{{5}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $+4\,f_{{1}}f_{{5}}a_{{0}}{a_{{2}}}^{
3}$ $-{f_{{2}}}^{2}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $-2\,f_{{2}}f_{{4}}a_{{0}}a_{{1}}a_{{2}}a_{{3}
}$ $+2\,f_{{2}}f_{{4}}a_{{0}}{a_{{2}}}^{3}$ $-8\,f_{{2}}f_{{5}}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $+6
\,f_{{2}}f_{{5}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $-4\,f_{{2}}f_{{6}}a_{{0}}a_{{2}}{a_{{3}}}^{2
}$ $-6\,f_{{2}}f_{{7}}a_{{0}}{a_{{3}}}^{3}$ $-{f_{{4}}}^{2}{a_{{1}}}^{2}{a_{{3}}}^{2}$ $+2\,{f_{
{4}}}^{2}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $-{f_{{4}}}^{2}{a_{{2}}}^{4}$ $-2\,f_{{4}}f_{{5}}{a_{{2
}}}^{3}a_{{3}}$ $+4\,f_{{4}}f_{{6}}{a_{{2}}}^{2}{a_{{3}}}^{2}$ $+6\,f_{{4}}f_{{7}}a_{{2}}{a_{
{3}}}^{3}$ $-4\,{f_{{5}}}^{2}{a_{{2}}}^{2}{a_{{3}}}^{2}$.

\noindent
${\bf eq[7]} =
8\,a_{{0}}a_{{3}}c_{{2}}c_{{3}}$ $-4\,a_{{1}}a_{{3}}c_{{1}}c_{{3}}$ $+2\,{a_{{2}}}^{2}c_{{1}}
c_{{3}}$ $-{a_{{2}}}^{2}{c_{{2}}}^{2}$ $+2\,a_{{2}}a_{{3}}c_{{0}}c_{{3}}$ $-2\,{a_{{3}}}^{2}c_{{0
}}c_{{2}}$ $+3\,{a_{{3}}}^{2}{c_{{1}}}^{2}
+
8\,f_{{0}}{a_{{0}}}^{2}a_{{1}}c_{{3}}$ $-8\,f_{{0}}{a_{{0}}}^{2}a_{{2}}c_{{2}}$ $-4\,f_{{0}}{
a_{{0}}}^{2}a_{{3}}c_{{1}}$ $+4\,f_{{0}}a_{{0}}{a_{{1}}}^{2}c_{{2}}$ $-4\,f_{{0}}a_{{0}}a_{{1
}}a_{{3}}c_{{0}}$ $-4\,f_{{0}}a_{{0}}{a_{{2}}}^{2}c_{{0}}$ $+4\,f_{{0}}{a_{{1}}}^{2}a_{{2}}c_
{{0}}$ $-2\,f_{{1}}{a_{{0}}}^{2}a_{{3}}c_{{2}}$ $+4\,f_{{1}}a_{{0}}a_{{1}}a_{{2}}c_{{2}}$ $+2\,f
_{{1}}a_{{0}}a_{{1}}a_{{3}}c_{{1}}$ $+4\,f_{{1}}a_{{0}}{a_{{2}}}^{2}c_{{1}}$ $-2\,f_{{1}}{a_{
{1}}}^{2}a_{{2}}c_{{1}}$ $+2\,f_{{1}}{a_{{1}}}^{2}a_{{3}}c_{{0}}$ $-2\,f_{{2}}a_{{0}}a_{{1}}a
_{{2}}c_{{3}}$ $-8\,f_{{2}}a_{{0}}a_{{1}}a_{{3}}c_{{2}}$ $+2\,f_{{2}}a_{{0}}{a_{{2}}}^{2}c_{{
2}}$ $-2\,f_{{2}}a_{{0}}{a_{{3}}}^{2}c_{{0}}$ $+2\,f_{{2}}{a_{{1}}}^{2}a_{{2}}c_{{2}}$ $-2\,f_{{
3}}a_{{0}}{a_{{2}}}^{2}c_{{3}}$ $-6\,f_{{3}}a_{{0}}a_{{2}}a_{{3}}c_{{2}}$ $-4\,f_{{3}}a_{{0}}
{a_{{3}}}^{2}c_{{1}}$ $+4\,f_{{3}}{a_{{1}}}^{2}a_{{3}}c_{{2}}$ $+6\,f_{{3}}a_{{1}}a_{{2}}a_{{
3}}c_{{1}}$ $+2\,f_{{3}}a_{{1}}{a_{{3}}}^{2}c_{{0}}$ $-2\,f_{{3}}{a_{{2}}}^{3}c_{{1}}$ $-2\,f_{{
3}}{a_{{2}}}^{2}a_{{3}}c_{{0}}$ $+6\,f_{{4}}a_{{0}}a_{{2}}a_{{3}}c_{{3}}$ $-2\,f_{{4}}a_{{0}}
{a_{{3}}}^{2}c_{{2}}$ $+4\,f_{{4}}{a_{{1}}}^{2}a_{{3}}c_{{3}}$ $-2\,f_{{4}}a_{{1}}{a_{{2}}}^{
2}c_{{3}}$ $-2\,f_{{4}}a_{{1}}{a_{{3}}}^{2}c_{{1}}$ $-2\,f_{{4}}{a_{{2}}}^{3}c_{{2}}$ $-2\,f_{{4
}}{a_{{2}}}^{2}a_{{3}}c_{{1}}$ $-4\,f_{{4}}a_{{2}}{a_{{3}}}^{2}c_{{0}}$ $+16\,f_{{5}}a_{{0}}{
a_{{3}}}^{2}c_{{3}}$ $+2\,f_{{5}}a_{{1}}a_{{2}}a_{{3}}c_{{3}}$ $+6\,f_{{5}}a_{{1}}{a_{{3}}}^{
2}c_{{2}}$ $-2\,f_{{5}}{a_{{2}}}^{3}c_{{3}}$ $-6\,f_{{5}}{a_{{2}}}^{2}a_{{3}}c_{{2}}$ $-4\,f_{{6
}}{a_{{2}}}^{2}a_{{3}}c_{{3}}$ $-4\,f_{{6}}{a_{{3}}}^{3}c_{{1}}$ $-8\,f_{{7}}a_{{2}}{a_{{3}}}
^{2}c_{{3}}$ $+4\,f_{{7}}{a_{{3}}}^{3}c_{{2}}
-16\,f_{{0}}f_{{5}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $+16\,f_{{0}}f_{{5}}a_{{0}}{a_{{1}}}^{2}a_
{{3}}$ $-4\,f_{{0}}f_{{5}}{a_{{1}}}^{3}a_{{2}}$ $+16\,f_{{0}}f_{{6}}a_{{0}}a_{{1}}a_{{2}}a_{{
3}}$ $-8\,f_{{0}}f_{{6}}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $+8\,f_{{0}}f_{{7}}a_{{0}}a_{{1}}{a_{{3}}
}^{2}$ $+16\,f_{{0}}f_{{7}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $-8\,f_{{0}}f_{{7}}{a_{{1}}}^{2}a_{{2
}}a_{{3}}$ $-4\,f_{{0}}f_{{7}}a_{{1}}{a_{{2}}}^{3}$ $+2\,{f_{{1}}}^{2}{a_{{0}}}^{3}a_{{2}}$ $-3
\,{f_{{1}}}^{2}{a_{{0}}}^{2}{a_{{1}}}^{2}$ $+2\,f_{{1}}f_{{2}}{a_{{0}}}^{3}a_{{3}}$ $-4\,f_{{
1}}f_{{2}}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $+2\,f_{{1}}f_{{3}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $-4\,f
_{{1}}f_{{3}}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $+2\,f_{{1}}f_{{3}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $+2
\,f_{{1}}f_{{4}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $-4\,f_{{1}}f_{{4}}a_{{0}}{a_{{1}}}^{2}a_{{3}
}$ $+2\,f_{{1}}f_{{4}}{a_{{1}}}^{3}a_{{2}}$ $-4\,f_{{1}}f_{{5}}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $+6\,
f_{{1}}f_{{5}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $-4\,f_{{1}}f_{{5}}a_{{0}}{a_{{2}}}^{3}$ $-2\,f_{
{1}}f_{{5}}{a_{{1}}}^{3}a_{{3}}$ $+2\,f_{{1}}f_{{5}}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $+4\,f_{{1}}f
_{{6}}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $-8\,f_{{1}}f_{{6}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $-4\,f_{{1
}}f_{{6}}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $+2\,f_{{1}}f_{{7}}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $-4\,f_
{{1}}f_{{7}}{a_{{1}}}^{2}{a_{{3}}}^{2}$ $-4\,f_{{1}}f_{{7}}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $-{f_
{{2}}}^{2}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $-2\,f_{{2}}f_{{3}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $+2\,f_
{{2}}f_{{3}}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $+2\,f_{{2}}f_{{4}}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $-6\,
f_{{2}}f_{{4}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $+2\,f_{{2}}f_{{4}}a_{{0}}{a_{{2}}}^{3}$ $+2\,f_{
{2}}f_{{4}}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $-22\,f_{{2}}f_{{5}}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $+6\,
f_{{2}}f_{{5}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $+6\,f_{{2}}f_{{5}}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $+
8\,f_{{2}}f_{{7}}a_{{0}}{a_{{3}}}^{3}$ $-2\,f_{{2}}f_{{7}}a_{{1}}a_{{2}}{a_{{3}}}^{2}$ $-2\,{
f_{{3}}}^{2}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $+2\,{f_{{3}}}^{2}a_{{0}}{a_{{2}}}^{3}$ $+2\,f_{{3}
}f_{{4}}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $-2\,f_{{3}}f_{{4}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $-2\,f_{
{3}}f_{{4}}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $+2\,f_{{3}}f_{{4}}a_{{1}}{a_{{2}}}^{3}$ $-12\,f_{{3}
}f_{{5}}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $+10\,f_{{3}}f_{{5}}{a_{{1}}}^{2}{a_{{3}}}^{2}$ $-4\,f_{
{3}}f_{{5}}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $+2\,f_{{3}}f_{{5}}{a_{{2}}}^{4}$ $-4\,f_{{3}}f_{{6}}
a_{{1}}a_{{2}}{a_{{3}}}^{2}$ $+4\,f_{{3}}f_{{6}}{a_{{2}}}^{3}a_{{3}}$ $-8\,f_{{3}}f_{{7}}a_{{
1}}{a_{{3}}}^{3}$ $+8\,f_{{3}}f_{{7}}{a_{{2}}}^{2}{a_{{3}}}^{2}$ $-{f_{{4}}}^{2}{a_{{1}}}^{2}
{a_{{3}}}^{2}$ $+2\,{f_{{4}}}^{2}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $-{f_{{4}}}^{2}{a_{{2}}}^{4}$ $-4
\,f_{{4}}f_{{5}}a_{{0}}{a_{{3}}}^{3}$ $+4\,f_{{4}}f_{{5}}a_{{1}}a_{{2}}{a_{{3}}}^{2}$ $-4\,f_
{{4}}f_{{5}}{a_{{2}}}^{3}a_{{3}}$ $+4\,f_{{4}}f_{{6}}a_{{1}}{a_{{3}}}^{3}$ $+4\,f_{{4}}f_{{6}
}{a_{{2}}}^{2}{a_{{3}}}^{2}$ $+12\,f_{{4}}f_{{7}}a_{{2}}{a_{{3}}}^{3}$ $+12\,{f_{{5}}}^{2}a_{
{1}}{a_{{3}}}^{3}$ $-12\,{f_{{5}}}^{2}{a_{{2}}}^{2}{a_{{3}}}^{2}$ $-8\,f_{{5}}f_{{6}}a_{{2}}{
a_{{3}}}^{3}$ $-4\,f_{{5}}f_{{7}}{a_{{3}}}^{4}$.

\noindent
${\bf eq[8]} =
-4\,{a_{{0}}}^{2}c_{{0}}c_{{2}}$ $+{a_{{0}}}^{2}{c_{{1}}}^{2}$ $-2\,a_{{0}}a_{{1}}c_{{0}}c_{{
1}}$ $-4\,a_{{0}}a_{{2}}{c_{{0}}}^{2}$ $+{a_{{1}}}^{2}{c_{{0}}}^{2}
-4\,f_{{5}}{a_{{0}}}^{2}a_{{2}}c_{{1}}$ $-12\,f_{{5}}{a_{{0}}}^{2}a_{{3}}c_{{0}}$ $+2\,f_{{5}
}a_{{0}}{a_{{1}}}^{2}c_{{1}}$ $+8\,f_{{5}}a_{{0}}a_{{1}}a_{{2}}c_{{0}}$ $-2\,f_{{5}}{a_{{1}}}
^{3}c_{{0}}$ $+8\,f_{{6}}{a_{{0}}}^{2}a_{{2}}c_{{2}}$ $-4\,f_{{6}}a_{{0}}{a_{{1}}}^{2}c_{{2}}
+12\,f_{{6}}a_{{0}}{a_{{2}}}^{2}c_{{0}}$ $-4\,f_{{6}}{a_{{1}}}^{2}a_{{2}}c_{{0}}$ $+28\,f_{{7
}}{a_{{0}}}^{2}a_{{2}}c_{{3}}$ $+12\,f_{{7}}{a_{{0}}}^{2}a_{{3}}c_{{2}}$ $-4\,f_{{7}}a_{{0}}{
a_{{1}}}^{2}c_{{3}}$ $-24\,f_{{7}}a_{{0}}a_{{1}}a_{{2}}c_{{2}}$ $+8\,f_{{7}}a_{{0}}a_{{1}}a_{
{3}}c_{{1}}$ $-26\,f_{{7}}a_{{0}}{a_{{2}}}^{2}c_{{1}}$ $-4\,f_{{7}}{a_{{1}}}^{2}a_{{3}}c_{{0}
}$ $-2\,f_{{7}}a_{{1}}{a_{{2}}}^{2}c_{{0}}
+
4\,{f_{{5}}}^{2}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $-4\,{f_{{5}}}^{2}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $+
{f_{{5}}}^{2}{a_{{1}}}^{4}$ $+24\,f_{{5}}f_{{6}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $-12\,f_{{5}}f_{
{6}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $-8\,f_{{5}}f_{{6}}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $+4\,f_{{5}}
f_{{6}}{a_{{1}}}^{3}a_{{2}}$ $+36\,f_{{5}}f_{{7}}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $-72\,f_{{5}}f_{
{7}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $+24\,f_{{5}}f_{{7}}a_{{0}}{a_{{2}}}^{3}$ $+4\,f_{{5}}f_{{7
}}{a_{{1}}}^{3}a_{{3}}$ $+2\,f_{{5}}f_{{7}}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $-8\,{f_{{6}}}^{2}a_{{0
}}{a_{{2}}}^{3}$ $+4\,{f_{{6}}}^{2}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $+12\,f_{{6}}f_{{7}}a_{{0}}{a_
{{2}}}^{2}a_{{3}}$ $+8\,f_{{6}}f_{{7}}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $+4\,f_{{6}}f_{{7}}a_{{1}}
{a_{{2}}}^{3}$ $+36\,{f_{{7}}}^{2}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $+4\,{f_{{7}}}^{2}{a_{{1}}}^{2
}{a_{{3}}}^{2}$ $+4\,{f_{{7}}}^{2}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $+{f_{{7}}}^{2}{a_{{2}}}^{4}$.

\noindent
${\bf eq[9]} =
a_{{0}}a_{{2}}c_{{2}}c_{{3}}$ $+2\,a_{{0}}a_{{3}}{c_{{2}}}^{2}$ $-a_{{1}}a_{{2}}{c_{{2}}}^{2}
-2\,a_{{1}}a_{{3}}c_{{1}}c_{{2}}$ $-a_{{2}}a_{{3}}{c_{{1}}}^{2}$ $-{a_{{3}}}^{2}c_{{0}}c_{{1}
}
+
4\,f_{{0}}{a_{{0}}}^{2}a_{{1}}c_{{2}}$ $+2\,f_{{0}}{a_{{0}}}^{2}a_{{2}}c_{{1}}$ $+2\,f_{{0}}a
_{{0}}a_{{1}}a_{{2}}c_{{0}}$ $-f_{{1}}{a_{{0}}}^{2}a_{{1}}c_{{3}}$ $+f_{{1}}{a_{{0}}}^{2}a_{{
2}}c_{{2}}$ $+f_{{1}}{a_{{0}}}^{2}a_{{3}}c_{{1}}$ $+f_{{1}}a_{{0}}{a_{{1}}}^{2}c_{{2}}$ $+2\,f_{
{1}}a_{{0}}a_{{1}}a_{{3}}c_{{0}}$ $-f_{{2}}{a_{{0}}}^{2}a_{{2}}c_{{3}}$ $-2\,f_{{2}}{a_{{0}}}
^{2}a_{{3}}c_{{2}}$ $+f_{{2}}a_{{0}}a_{{1}}a_{{2}}c_{{2}}$ $+f_{{3}}{a_{{0}}}^{2}a_{{3}}c_{{3
}}$ $+f_{{3}}a_{{0}}a_{{1}}a_{{3}}c_{{2}}$ $-f_{{3}}a_{{0}}{a_{{2}}}^{2}c_{{2}}$ $-f_{{3}}a_{{0}
}{a_{{3}}}^{2}c_{{0}}$ $-f_{{4}}a_{{0}}{a_{{2}}}^{2}c_{{3}}$ $+3\,f_{{4}}a_{{0}}a_{{2}}a_{{3}
}c_{{2}}$ $+f_{{4}}a_{{0}}{a_{{3}}}^{2}c_{{1}}$ $+2\,f_{{4}}{a_{{1}}}^{2}a_{{3}}c_{{2}}$ $+f_{{4
}}a_{{1}}{a_{{3}}}^{2}c_{{0}}$ $+2\,f_{{4}}{a_{{2}}}^{3}c_{{1}}$ $+2\,f_{{4}}{a_{{2}}}^{2}a_{
{3}}c_{{0}}$ $-f_{{5}}a_{{0}}a_{{2}}a_{{3}}c_{{3}}$ $+10\,f_{{5}}a_{{0}}{a_{{3}}}^{2}c_{{2}}$ $-
5\,f_{{5}}a_{{1}}{a_{{3}}}^{2}c_{{1}}$ $+6\,f_{{5}}{a_{{2}}}^{2}a_{{3}}c_{{1}}$ $+5\,f_{{5}}a
_{{2}}{a_{{3}}}^{2}c_{{0}}$ $+2\,f_{{6}}a_{{2}}{a_{{3}}}^{2}c_{{1}}$ $+f_{{7}}a_{{2}}{a_{{3}}
}^{2}c_{{2}}$ $+4\,f_{{7}}{a_{{3}}}^{3}c_{{1}}
+
12\,f_{{0}}f_{{5}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $-4\,f_{{0}}f_{{5}}{a_{{0}}}^{2}{a_{{2}}}^{
2}$ $-2\,f_{{0}}f_{{5}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $-4\,f_{{0}}f_{{6}}a_{{0}}a_{{1}}{a_{{2}}
}^{2}$ $-8\,f_{{0}}f_{{7}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $+2\,f_{{0}}f_{{7}}a_{{0}}{a_{{2}}}^{
3}$ $-{f_{{1}}}^{2}{a_{{0}}}^{3}a_{{1}}$ $-f_{{1}}f_{{2}}{a_{{0}}}^{3}a_{{2}}$ $+f_{{1}}f_{{3}}{
a_{{0}}}^{2}a_{{1}}a_{{2}}$ $-2\,f_{{1}}f_{{4}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $+f_{{1}}f_{{4}}{
a_{{0}}}^{2}{a_{{2}}}^{2}$ $+f_{{1}}f_{{4}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $+f_{{1}}f_{{5}}a_{{0
}}{a_{{1}}}^{2}a_{{3}}$ $-4\,f_{{1}}f_{{6}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $-7\,f_{{1}}f_{{7}}a
_{{0}}a_{{1}}{a_{{3}}}^{2}$ $+2\,f_{{1}}f_{{7}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $+f_{{2}}f_{{3}}{
a_{{0}}}^{2}{a_{{2}}}^{2}$ $-f_{{2}}f_{{4}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $+f_{{2}}f_{{4}}a_{{0
}}a_{{1}}{a_{{2}}}^{2}$ $-6\,f_{{2}}f_{{5}}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $+3\,f_{{2}}f_{{5}}a_{
{0}}a_{{1}}a_{{2}}a_{{3}}$ $+f_{{2}}f_{{7}}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $+f_{{3}}f_{{4}}a_{{0
}}a_{{1}}a_{{2}}a_{{3}}$ $-f_{{3}}f_{{4}}a_{{0}}{a_{{2}}}^{3}$ $+4\,f_{{3}}f_{{5}}a_{{0}}a_{{
1}}{a_{{3}}}^{2}$ $-3\,f_{{3}}f_{{5}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $+2\,f_{{3}}f_{{6}}a_{{0}}a
_{{2}}{a_{{3}}}^{2}$ $+3\,f_{{3}}f_{{7}}a_{{0}}{a_{{3}}}^{3}$ $-{f_{{4}}}^{2}a_{{0}}a_{{1}}{a
_{{3}}}^{2}$ $+{f_{{4}}}^{2}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $+{f_{{4}}}^{2}{a_{{1}}}^{2}a_{{2}}a
_{{3}}$ $-{f_{{4}}}^{2}a_{{1}}{a_{{2}}}^{3}$ $+6\,f_{{4}}f_{{5}}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $+5
\,f_{{4}}f_{{5}}{a_{{1}}}^{2}{a_{{3}}}^{2}$ $-2\,f_{{4}}f_{{5}}a_{{1}}{a_{{2}}}^{2}a_{{3}}
-2\,f_{{4}}f_{{5}}{a_{{2}}}^{4}$ $-2\,f_{{4}}f_{{6}}a_{{1}}a_{{2}}{a_{{3}}}^{2}$ $-4\,f_{{4}}
f_{{6}}{a_{{2}}}^{3}a_{{3}}$ $-4\,f_{{4}}f_{{7}}a_{{1}}{a_{{3}}}^{3}$ $-5\,f_{{4}}f_{{7}}{a_{
{2}}}^{2}{a_{{3}}}^{2}$ $+12\,{f_{{5}}}^{2}a_{{0}}{a_{{3}}}^{3}$ $+4\,{f_{{5}}}^{2}a_{{1}}a_{
{2}}{a_{{3}}}^{2}$ $-5\,{f_{{5}}}^{2}{a_{{2}}}^{3}a_{{3}}$ $-10\,f_{{5}}f_{{6}}{a_{{2}}}^{2}{
a_{{3}}}^{2}$ $-13\,f_{{5}}f_{{7}}a_{{2}}{a_{{3}}}^{3}$.

\noindent
${\bf eq[10]} =
-a_{{0}}a_{{1}}{c_{{3}}}^{2}$ $+3\,a_{{0}}a_{{3}}{c_{{2}}}^{2}$ $+{a_{{1}}}^{2}c_{{2}}c_{{3}}
-a_{{1}}a_{{2}}{c_{{2}}}^{2}$ $-a_{{1}}a_{{3}}c_{{0}}c_{{3}}$ $+a_{{1}}a_{{3}}c_{{1}}c_{{2}}$ $+
2\,a_{{2}}a_{{3}}c_{{0}}c_{{2}}$ $+a_{{2}}a_{{3}}{c_{{1}}}^{2}$ $+3\,{a_{{3}}}^{2}c_{{0}}c_{{
1}}
+
4\,f_{{0}}{a_{{0}}}^{2}a_{{1}}c_{{2}}$ $-4\,f_{{0}}{a_{{0}}}^{2}a_{{3}}c_{{0}}$ $+2\,f_{{0}}a
_{{0}}{a_{{1}}}^{2}c_{{1}}$ $+8\,f_{{0}}a_{{0}}a_{{1}}a_{{2}}c_{{0}}$ $-2\,f_{{0}}{a_{{1}}}^{
3}c_{{0}}$ $-2\,f_{{1}}{a_{{0}}}^{2}a_{{1}}c_{{3}}$ $+f_{{1}}{a_{{0}}}^{2}a_{{3}}c_{{1}}$ $+3\,f
_{{1}}a_{{0}}{a_{{1}}}^{2}c_{{2}}$ $+5\,f_{{1}}a_{{0}}a_{{1}}a_{{3}}c_{{0}}$ $+f_{{1}}{a_{{1}
}}^{3}c_{{1}}$ $-2\,f_{{2}}{a_{{0}}}^{2}a_{{2}}c_{{3}}$ $-4\,f_{{2}}{a_{{0}}}^{2}a_{{3}}c_{{2
}}$ $+4\,f_{{2}}a_{{0}}a_{{1}}a_{{2}}c_{{2}}$ $-3\,f_{{2}}a_{{0}}a_{{1}}a_{{3}}c_{{1}}$ $+2\,f_{
{2}}a_{{0}}{a_{{2}}}^{2}c_{{1}}$ $+f_{{2}}{a_{{1}}}^{2}a_{{2}}c_{{1}}$ $+2\,f_{{2}}{a_{{1}}}^
{2}a_{{3}}c_{{0}}$ $-f_{{3}}{a_{{0}}}^{2}a_{{3}}c_{{3}}$ $+2\,f_{{3}}a_{{0}}a_{{1}}a_{{2}}c_{
{3}}$ $+f_{{3}}a_{{0}}a_{{1}}a_{{3}}c_{{2}}$ $-3\,f_{{3}}a_{{0}}{a_{{3}}}^{2}c_{{0}}$ $-f_{{3}}{
a_{{1}}}^{2}a_{{2}}c_{{2}}$ $-f_{{3}}{a_{{1}}}^{2}a_{{3}}c_{{1}}$ $+f_{{3}}a_{{1}}a_{{2}}a_{{
3}}c_{{0}}$ $-2\,f_{{4}}a_{{0}}{a_{{2}}}^{2}c_{{3}}$ $+4\,f_{{4}}a_{{0}}a_{{2}}a_{{3}}c_{{2}}
+f_{{4}}a_{{0}}{a_{{3}}}^{2}c_{{1}}$ $+f_{{4}}{a_{{1}}}^{2}a_{{2}}c_{{3}}$ $+f_{{4}}{a_{{1}}}
^{2}a_{{3}}c_{{2}}$ $-f_{{4}}a_{{1}}{a_{{3}}}^{2}c_{{0}}$ $+2\,f_{{4}}{a_{{2}}}^{3}c_{{1}}$ $+4
\,f_{{4}}{a_{{2}}}^{2}a_{{3}}c_{{0}}$ $-8\,f_{{5}}a_{{0}}a_{{2}}a_{{3}}c_{{3}}$ $+14\,f_{{5}}
a_{{0}}{a_{{3}}}^{2}c_{{2}}$ $+4\,f_{{5}}{a_{{1}}}^{2}a_{{3}}c_{{3}}$ $+f_{{5}}a_{{1}}{a_{{3}
}}^{2}c_{{1}}$ $+5\,f_{{5}}{a_{{2}}}^{2}a_{{3}}c_{{1}}$ $+10\,f_{{5}}a_{{2}}{a_{{3}}}^{2}c_{{0
}}$ $-4\,f_{{6}}a_{{0}}{a_{{3}}}^{2}c_{{3}}$ $+2\,f_{{6}}a_{{1}}a_{{2}}a_{{3}}c_{{3}}$ $-2\,f_{{
6}}{a_{{2}}}^{2}a_{{3}}c_{{2}}$ $-4\,f_{{6}}a_{{2}}{a_{{3}}}^{2}c_{{1}}$ $+f_{{7}}a_{{1}}{a_{
{3}}}^{2}c_{{3}}$ $+f_{{7}}{a_{{2}}}^{2}a_{{3}}c_{{3}}$ $-4\,f_{{7}}a_{{2}}{a_{{3}}}^{2}c_{{2
}}$ $-8\,f_{{7}}{a_{{3}}}^{3}c_{{1}}
+
12\,f_{{0}}f_{{5}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $-8\,f_{{0}}f_{{5}}a_{{0}}{a_{{1}}}^{2}a_{{
2}}$ $+2\,f_{{0}}f_{{5}}{a_{{1}}}^{4}$ $+8\,f_{{0}}f_{{6}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $-4\,f_{{0
}}f_{{6}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $-12\,f_{{0}}f_{{6}}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $+4\,f
_{{0}}f_{{6}}{a_{{1}}}^{3}a_{{2}}$ $+12\,f_{{0}}f_{{7}}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $-24\,f_{{0
}}f_{{7}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $+4\,f_{{0}}f_{{7}}{a_{{1}}}^{3}a_{{3}}$ $+2\,f_{{0}}f
_{{7}}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $-{f_{{1}}}^{2}{a_{{0}}}^{3}a_{{1}}$ $-f_{{1}}f_{{2}}{a_{{0
}}}^{2}{a_{{1}}}^{2}$ $+2\,f_{{1}}f_{{3}}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $-f_{{1}}f_{{3}}a_{{0}}
{a_{{1}}}^{3}$ $-2\,f_{{1}}f_{{4}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $+3\,f_{{1}}f_{{4}}a_{{0}}{a_{
{1}}}^{2}a_{{2}}$ $-f_{{1}}f_{{4}}{a_{{1}}}^{4}$ $-2\,f_{{1}}f_{{5}}{a_{{0}}}^{2}a_{{2}}a_{{3
}}$ $+5\,f_{{1}}f_{{5}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $-f_{{1}}f_{{5}}{a_{{1}}}^{3}a_{{2}}$ $-8\,f
_{{1}}f_{{6}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $-16\,f_{{1}}f_{{7}}a_{{0}}a_{{1}}{a_{{3}}}^{2}
+f_{{1}}f_{{7}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $+f_{{1}}f_{{7}}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $+{f
_{{2}}}^{2}{a_{{0}}}^{3}a_{{3}}$ $-{f_{{2}}}^{2}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $+2\,f_{{2}}f_{{
3}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $-f_{{2}}f_{{3}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $-2\,f_{{2}}f_{{
4}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $+f_{{2}}f_{{4}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $+2\,f_{{2}}f_{{
4}}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $-f_{{2}}f_{{4}}{a_{{1}}}^{3}a_{{2}}$ $-10\,f_{{2}}f_{{5}}{a_
{{0}}}^{2}{a_{{3}}}^{2}$ $+14\,f_{{2}}f_{{5}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $-2\,f_{{2}}f_{{5}
}a_{{0}}{a_{{2}}}^{3}$ $-2\,f_{{2}}f_{{5}}{a_{{1}}}^{3}a_{{3}}$ $-f_{{2}}f_{{5}}{a_{{1}}}^{2}
{a_{{2}}}^{2}$ $+4\,f_{{2}}f_{{6}}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $-2\,f_{{2}}f_{{6}}a_{{0}}{a_{
{2}}}^{2}a_{{3}}$ $-4\,f_{{2}}f_{{6}}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $-4\,f_{{2}}f_{{7}}{a_{{1}}
}^{2}{a_{{3}}}^{2}$ $-f_{{2}}f_{{7}}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $+{f_{{3}}}^{2}a_{{0}}{a_{{1
}}}^{2}a_{{3}}$ $-{f_{{3}}}^{2}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $+f_{{3}}f_{{4}}{a_{{1}}}^{3}a_{{
3}}$ $-f_{{3}}f_{{4}}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $+4\,f_{{3}}f_{{5}}a_{{0}}a_{{1}}{a_{{3}}}^{
2}$ $+f_{{3}}f_{{5}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $-3\,f_{{3}}f_{{5}}{a_{{1}}}^{2}a_{{2}}a_{{3
}}$ $+6\,f_{{3}}f_{{6}}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $-2\,f_{{3}}f_{{6}}a_{{1}}{a_{{2}}}^{2}a_
{{3}}$ $+9\,f_{{3}}f_{{7}}a_{{0}}{a_{{3}}}^{3}$ $-2\,f_{{3}}f_{{7}}a_{{1}}a_{{2}}{a_{{3}}}^{2
}$ $-f_{{3}}f_{{7}}{a_{{2}}}^{3}a_{{3}}$ $-{f_{{4}}}^{2}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $+{f_{{4}}}
^{2}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $+{f_{{4}}}^{2}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $-{f_{{4}}}^{2}a
_{{1}}{a_{{2}}}^{3}$ $+8\,f_{{4}}f_{{5}}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $+3\,f_{{4}}f_{{5}}{a_{{
1}}}^{2}{a_{{3}}}^{2}$ $-2\,f_{{4}}f_{{5}}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $-2\,f_{{4}}f_{{5}}{a_
{{2}}}^{4}$ $-6\,f_{{4}}f_{{6}}{a_{{2}}}^{3}a_{{3}}$ $+2\,f_{{4}}f_{{7}}a_{{1}}{a_{{3}}}^{3}$ $-
11\,f_{{4}}f_{{7}}{a_{{2}}}^{2}{a_{{3}}}^{2}$ $+16\,{f_{{5}}}^{2}a_{{0}}{a_{{3}}}^{3}$ $+4\,{
f_{{5}}}^{2}a_{{1}}a_{{2}}{a_{{3}}}^{2}$ $-6\,{f_{{5}}}^{2}{a_{{2}}}^{3}a_{{3}}$ $-16\,f_{{5}
}f_{{6}}{a_{{2}}}^{2}{a_{{3}}}^{2}$ $-26\,f_{{5}}f_{{7}}a_{{2}}{a_{{3}}}^{3}$.

\noindent
${\bf eq[11]} =
2\,a_{{0}}a_{{3}}{c_{{2}}}^{2}$ $-2\,a_{{1}}a_{{3}}c_{{1}}c_{{2}}$ $+{a_{{2}}}^{2}c_{{1}}c_{{
2}}$ $+a_{{2}}a_{{3}}c_{{0}}c_{{2}}$ $-a_{{2}}a_{{3}}{c_{{1}}}^{2}$ $-{a_{{3}}}^{2}c_{{0}}c_{{1}
}
+
4\,f_{{0}}{a_{{0}}}^{2}a_{{1}}c_{{2}}$ $+2\,f_{{0}}{a_{{0}}}^{2}a_{{2}}c_{{1}}$ $+2\,f_{{0}}a
_{{0}}a_{{1}}a_{{2}}c_{{0}}$ $-f_{{1}}{a_{{0}}}^{2}a_{{1}}c_{{3}}$ $+f_{{1}}{a_{{0}}}^{2}a_{{
2}}c_{{2}}$ $+f_{{1}}{a_{{0}}}^{2}a_{{3}}c_{{1}}$ $+f_{{1}}a_{{0}}{a_{{1}}}^{2}c_{{2}}$ $+2\,f_{
{1}}a_{{0}}a_{{1}}a_{{3}}c_{{0}}$ $-f_{{2}}{a_{{0}}}^{2}a_{{2}}c_{{3}}$ $-2\,f_{{2}}{a_{{0}}}
^{2}a_{{3}}c_{{2}}$ $+f_{{2}}a_{{0}}a_{{1}}a_{{2}}c_{{2}}$ $+f_{{3}}{a_{{0}}}^{2}a_{{3}}c_{{3
}}$ $+f_{{3}}a_{{0}}a_{{1}}a_{{3}}c_{{2}}$ $-f_{{3}}a_{{0}}{a_{{2}}}^{2}c_{{2}}$ $-f_{{3}}a_{{0}
}{a_{{3}}}^{2}c_{{0}}$ $-f_{{4}}a_{{0}}{a_{{2}}}^{2}c_{{3}}$ $+3\,f_{{4}}a_{{0}}a_{{2}}a_{{3}
}c_{{2}}$ $+f_{{4}}a_{{0}}{a_{{3}}}^{2}c_{{1}}$ $+2\,f_{{4}}{a_{{1}}}^{2}a_{{3}}c_{{2}}$ $+f_{{4
}}a_{{1}}{a_{{3}}}^{2}c_{{0}}$ $+2\,f_{{4}}{a_{{2}}}^{3}c_{{1}}$ $+2\,f_{{4}}{a_{{2}}}^{2}a_{
{3}}c_{{0}}$ $+f_{{5}}a_{{0}}a_{{2}}a_{{3}}c_{{3}}$ $+10\,f_{{5}}a_{{0}}{a_{{3}}}^{2}c_{{2}}$ $-
5\,f_{{5}}a_{{1}}{a_{{3}}}^{2}c_{{1}}$ $-f_{{5}}{a_{{2}}}^{3}c_{{2}}$ $+4\,f_{{5}}{a_{{2}}}^{
2}a_{{3}}c_{{1}}$ $+3\,f_{{5}}a_{{2}}{a_{{3}}}^{2}c_{{0}}$ $-2\,f_{{6}}{a_{{2}}}^{2}a_{{3}}c_
{{2}}$ $+2\,f_{{6}}a_{{2}}{a_{{3}}}^{2}c_{{1}}$ $-2\,f_{{7}}a_{{2}}{a_{{3}}}^{2}c_{{2}}$ $+4\,f_
{{7}}{a_{{3}}}^{3}c_{{1}}
+
12\,f_{{0}}f_{{5}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $-4\,f_{{0}}f_{{5}}{a_{{0}}}^{2}{a_{{2}}}^{
2}$ $-2\,f_{{0}}f_{{5}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $-4\,f_{{0}}f_{{6}}a_{{0}}a_{{1}}{a_{{2}}
}^{2}$ $-8\,f_{{0}}f_{{7}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $+2\,f_{{0}}f_{{7}}a_{{0}}{a_{{2}}}^{
3}$ $-{f_{{1}}}^{2}{a_{{0}}}^{3}a_{{1}}$ $-f_{{1}}f_{{2}}{a_{{0}}}^{3}a_{{2}}$ $+f_{{1}}f_{{3}}{
a_{{0}}}^{2}a_{{1}}a_{{2}}$ $-2\,f_{{1}}f_{{4}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $+f_{{1}}f_{{4}}{
a_{{0}}}^{2}{a_{{2}}}^{2}$ $+f_{{1}}f_{{4}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $+f_{{1}}f_{{5}}a_{{0
}}{a_{{1}}}^{2}a_{{3}}$ $-4\,f_{{1}}f_{{6}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $-7\,f_{{1}}f_{{7}}a
_{{0}}a_{{1}}{a_{{3}}}^{2}$ $+2\,f_{{1}}f_{{7}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $+f_{{2}}f_{{3}}{
a_{{0}}}^{2}{a_{{2}}}^{2}$ $-f_{{2}}f_{{4}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $+f_{{2}}f_{{4}}a_{{0
}}a_{{1}}{a_{{2}}}^{2}$ $-6\,f_{{2}}f_{{5}}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $+3\,f_{{2}}f_{{5}}a_{
{0}}a_{{1}}a_{{2}}a_{{3}}$ $+f_{{2}}f_{{7}}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $+f_{{3}}f_{{4}}a_{{0
}}a_{{1}}a_{{2}}a_{{3}}$ $-f_{{3}}f_{{4}}a_{{0}}{a_{{2}}}^{3}$ $+4\,f_{{3}}f_{{5}}a_{{0}}a_{{
1}}{a_{{3}}}^{2}$ $-3\,f_{{3}}f_{{5}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $+2\,f_{{3}}f_{{6}}a_{{0}}a
_{{2}}{a_{{3}}}^{2}$ $+3\,f_{{3}}f_{{7}}a_{{0}}{a_{{3}}}^{3}$ $-{f_{{4}}}^{2}a_{{0}}a_{{1}}{a
_{{3}}}^{2}$ $+{f_{{4}}}^{2}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $+{f_{{4}}}^{2}{a_{{1}}}^{2}a_{{2}}a
_{{3}}$ $-{f_{{4}}}^{2}a_{{1}}{a_{{2}}}^{3}$ $+6\,f_{{4}}f_{{5}}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $+5
\,f_{{4}}f_{{5}}{a_{{1}}}^{2}{a_{{3}}}^{2}$ $-2\,f_{{4}}f_{{5}}a_{{1}}{a_{{2}}}^{2}a_{{3}}
-2\,f_{{4}}f_{{5}}{a_{{2}}}^{4}$ $-2\,f_{{4}}f_{{6}}a_{{1}}a_{{2}}{a_{{3}}}^{2}$ $-4\,f_{{4}}
f_{{6}}{a_{{2}}}^{3}a_{{3}}$ $-4\,f_{{4}}f_{{7}}a_{{1}}{a_{{3}}}^{3}$ $-5\,f_{{4}}f_{{7}}{a_{
{2}}}^{2}{a_{{3}}}^{2}$ $+12\,{f_{{5}}}^{2}a_{{0}}{a_{{3}}}^{3}$ $-3\,{f_{{5}}}^{2}{a_{{2}}}^
{3}a_{{3}}$ $-6\,f_{{5}}f_{{6}}{a_{{2}}}^{2}{a_{{3}}}^{2}$ $-7\,f_{{5}}f_{{7}}a_{{2}}{a_{{3}}
}^{3}$.

\noindent
${\bf eq[12]} =
-3\,a_{{0}}a_{{3}}{c_{{2}}}^{2}$ $+a_{{1}}a_{{2}}c_{{1}}c_{{3}}$ $+a_{{1}}a_{{2}}{c_{{2}}}^{2
}$ $+2\,a_{{1}}a_{{3}}c_{{0}}c_{{3}}$ $-a_{{1}}a_{{3}}c_{{1}}c_{{2}}$ $-2\,a_{{2}}a_{{3}}c_{{0}}
c_{{2}}$ $-a_{{2}}a_{{3}}{c_{{1}}}^{2}$ $-3\,{a_{{3}}}^{2}c_{{0}}c_{{1}}
-4\,f_{{0}}{a_{{0}}}^{2}a_{{1}}c_{{2}}$ $+4\,f_{{0}}{a_{{0}}}^{2}a_{{3}}c_{{0}}$ $-2\,f_{{0}}
a_{{0}}{a_{{1}}}^{2}c_{{1}}$ $-8\,f_{{0}}a_{{0}}a_{{1}}a_{{2}}c_{{0}}$ $+2\,f_{{0}}{a_{{1}}}^
{3}c_{{0}}$ $+2\,f_{{1}}{a_{{0}}}^{2}a_{{1}}c_{{3}}$ $-f_{{1}}{a_{{0}}}^{2}a_{{3}}c_{{1}}$ $-3\,
f_{{1}}a_{{0}}{a_{{1}}}^{2}c_{{2}}$ $-5\,f_{{1}}a_{{0}}a_{{1}}a_{{3}}c_{{0}}$ $-f_{{1}}{a_{{1
}}}^{3}c_{{1}}$ $+2\,f_{{2}}{a_{{0}}}^{2}a_{{2}}c_{{3}}$ $+4\,f_{{2}}{a_{{0}}}^{2}a_{{3}}c_{{
2}}$ $-4\,f_{{2}}a_{{0}}a_{{1}}a_{{2}}c_{{2}}$ $+3\,f_{{2}}a_{{0}}a_{{1}}a_{{3}}c_{{1}}$ $-2\,f_
{{2}}a_{{0}}{a_{{2}}}^{2}c_{{1}}$ $-f_{{2}}{a_{{1}}}^{2}a_{{2}}c_{{1}}$ $-2\,f_{{2}}{a_{{1}}}
^{2}a_{{3}}c_{{0}}$ $+f_{{3}}{a_{{0}}}^{2}a_{{3}}c_{{3}}$ $-2\,f_{{3}}a_{{0}}a_{{1}}a_{{2}}c_
{{3}}$ $-f_{{3}}a_{{0}}a_{{1}}a_{{3}}c_{{2}}$ $+3\,f_{{3}}a_{{0}}{a_{{3}}}^{2}c_{{0}}$ $+f_{{3}}
{a_{{1}}}^{2}a_{{2}}c_{{2}}$ $+f_{{3}}{a_{{1}}}^{2}a_{{3}}c_{{1}}$ $-f_{{3}}a_{{1}}a_{{2}}a_{
{3}}c_{{0}}$ $+2\,f_{{4}}a_{{0}}{a_{{2}}}^{2}c_{{3}}$ $-4\,f_{{4}}a_{{0}}a_{{2}}a_{{3}}c_{{2}
}$ $-f_{{4}}a_{{0}}{a_{{3}}}^{2}c_{{1}}$ $-f_{{4}}{a_{{1}}}^{2}a_{{2}}c_{{3}}$ $-f_{{4}}{a_{{1}}
}^{2}a_{{3}}c_{{2}}$ $+f_{{4}}a_{{1}}{a_{{3}}}^{2}c_{{0}}$ $-2\,f_{{4}}{a_{{2}}}^{3}c_{{1}}$ $-4
\,f_{{4}}{a_{{2}}}^{2}a_{{3}}c_{{0}}$ $+8\,f_{{5}}a_{{0}}a_{{2}}a_{{3}}c_{{3}}$ $-14\,f_{{5}}
a_{{0}}{a_{{3}}}^{2}c_{{2}}$ $-2\,f_{{5}}{a_{{1}}}^{2}a_{{3}}c_{{3}}$ $-f_{{5}}a_{{1}}{a_{{2}
}}^{2}c_{{3}}$ $-f_{{5}}a_{{1}}{a_{{3}}}^{2}c_{{1}}$ $-5\,f_{{5}}{a_{{2}}}^{2}a_{{3}}c_{{1}}$ $-
10\,f_{{5}}a_{{2}}{a_{{3}}}^{2}c_{{0}}$ $+4\,f_{{6}}a_{{0}}{a_{{3}}}^{2}c_{{3}}$ $-4\,f_{{6}}
a_{{1}}a_{{2}}a_{{3}}c_{{3}}$ $+2\,f_{{6}}{a_{{2}}}^{2}a_{{3}}c_{{2}}$ $+4\,f_{{6}}a_{{2}}{a_
{{3}}}^{2}c_{{1}}$ $-4\,f_{{7}}a_{{1}}{a_{{3}}}^{2}c_{{3}}$ $-f_{{7}}{a_{{2}}}^{2}a_{{3}}c_{{
3}}$ $+4\,f_{{7}}a_{{2}}{a_{{3}}}^{2}c_{{2}}$ $+8\,f_{{7}}{a_{{3}}}^{3}c_{{1}}
-12\,f_{{0}}f_{{5}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $+8\,f_{{0}}f_{{5}}a_{{0}}{a_{{1}}}^{2}a_{
{2}}$ $-2\,f_{{0}}f_{{5}}{a_{{1}}}^{4}$ $-8\,f_{{0}}f_{{6}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $+4\,f_{
{0}}f_{{6}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $+12\,f_{{0}}f_{{6}}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $-4
\,f_{{0}}f_{{6}}{a_{{1}}}^{3}a_{{2}}$ $-12\,f_{{0}}f_{{7}}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $+24\,f
_{{0}}f_{{7}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $-4\,f_{{0}}f_{{7}}{a_{{1}}}^{3}a_{{3}}$ $-2\,f_{{0
}}f_{{7}}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $+{f_{{1}}}^{2}{a_{{0}}}^{3}a_{{1}}$ $+f_{{1}}f_{{2}}{a_
{{0}}}^{2}{a_{{1}}}^{2}$ $-2\,f_{{1}}f_{{3}}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $+f_{{1}}f_{{3}}a_{{0
}}{a_{{1}}}^{3}$ $+2\,f_{{1}}f_{{4}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $-3\,f_{{1}}f_{{4}}a_{{0}}{a
_{{1}}}^{2}a_{{2}}$ $+f_{{1}}f_{{4}}{a_{{1}}}^{4}$ $+2\,f_{{1}}f_{{5}}{a_{{0}}}^{2}a_{{2}}a_{
{3}}$ $-5\,f_{{1}}f_{{5}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $+f_{{1}}f_{{5}}{a_{{1}}}^{3}a_{{2}}$ $+8
\,f_{{1}}f_{{6}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $+16\,f_{{1}}f_{{7}}a_{{0}}a_{{1}}{a_{{3}}}^
{2}$ $-f_{{1}}f_{{7}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $-f_{{1}}f_{{7}}{a_{{1}}}^{2}a_{{2}}a_{{3}}
-{f_{{2}}}^{2}{a_{{0}}}^{3}a_{{3}}$ $+{f_{{2}}}^{2}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $-2\,f_{{2}}f
_{{3}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $+f_{{2}}f_{{3}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $+2\,f_{{2}}f
_{{4}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $-f_{{2}}f_{{4}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $-2\,f_{{2}}f
_{{4}}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $+f_{{2}}f_{{4}}{a_{{1}}}^{3}a_{{2}}$ $+10\,f_{{2}}f_{{5}}
{a_{{0}}}^{2}{a_{{3}}}^{2}$ $-14\,f_{{2}}f_{{5}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $+2\,f_{{2}}f_{
{5}}a_{{0}}{a_{{2}}}^{3}$ $+2\,f_{{2}}f_{{5}}{a_{{1}}}^{3}a_{{3}}$ $+f_{{2}}f_{{5}}{a_{{1}}}^
{2}{a_{{2}}}^{2}$ $-4\,f_{{2}}f_{{6}}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $+2\,f_{{2}}f_{{6}}a_{{0}}{
a_{{2}}}^{2}a_{{3}}$ $+4\,f_{{2}}f_{{6}}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $+4\,f_{{2}}f_{{7}}{a_{{
1}}}^{2}{a_{{3}}}^{2}$ $+f_{{2}}f_{{7}}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $-{f_{{3}}}^{2}a_{{0}}{a_
{{1}}}^{2}a_{{3}}$ $+{f_{{3}}}^{2}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $-f_{{3}}f_{{4}}{a_{{1}}}^{3}a
_{{3}}$ $+f_{{3}}f_{{4}}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $-4\,f_{{3}}f_{{5}}a_{{0}}a_{{1}}{a_{{3}}
}^{2}$ $-f_{{3}}f_{{5}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $+3\,f_{{3}}f_{{5}}{a_{{1}}}^{2}a_{{2}}a_
{{3}}$ $-6\,f_{{3}}f_{{6}}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $+2\,f_{{3}}f_{{6}}a_{{1}}{a_{{2}}}^{2
}a_{{3}}$ $-9\,f_{{3}}f_{{7}}a_{{0}}{a_{{3}}}^{3}$ $+2\,f_{{3}}f_{{7}}a_{{1}}a_{{2}}{a_{{3}}}
^{2}$ $+f_{{3}}f_{{7}}{a_{{2}}}^{3}a_{{3}}$ $+{f_{{4}}}^{2}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $-{f_{{4
}}}^{2}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $-{f_{{4}}}^{2}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $+{f_{{4}}}^{
2}a_{{1}}{a_{{2}}}^{3}$ $-8\,f_{{4}}f_{{5}}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $-3\,f_{{4}}f_{{5}}{a
_{{1}}}^{2}{a_{{3}}}^{2}$ $+2\,f_{{4}}f_{{5}}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $+2\,f_{{4}}f_{{5}}
{a_{{2}}}^{4}$ $+6\,f_{{4}}f_{{6}}{a_{{2}}}^{3}a_{{3}}$ $-2\,f_{{4}}f_{{7}}a_{{1}}{a_{{3}}}^{
3}$ $+11\,f_{{4}}f_{{7}}{a_{{2}}}^{2}{a_{{3}}}^{2}$ $-16\,{f_{{5}}}^{2}a_{{0}}{a_{{3}}}^{3}$ $-4
\,{f_{{5}}}^{2}a_{{1}}a_{{2}}{a_{{3}}}^{2}$ $+6\,{f_{{5}}}^{2}{a_{{2}}}^{3}a_{{3}}$ $+16\,f_{
{5}}f_{{6}}{a_{{2}}}^{2}{a_{{3}}}^{2}$ $+26\,f_{{5}}f_{{7}}a_{{2}}{a_{{3}}}^{3}$.

\noindent
${\bf eq[13]} =
-a_{{0}}a_{{1}}c_{{2}}c_{{3}}$ $+a_{{0}}a_{{2}}c_{{1}}c_{{3}}$ $+a_{{0}}a_{{3}}c_{{0}}c_{{3}}
+4\,a_{{0}}a_{{3}}c_{{1}}c_{{2}}$ $+{a_{{1}}}^{2}{c_{{2}}}^{2}
-4\,f_{{0}}{a_{{0}}}^{3}c_{{2}}$ $-4\,f_{{0}}{a_{{0}}}^{2}a_{{2}}c_{{0}}$ $+2\,f_{{1}}{a_{{0}
}}^{2}a_{{2}}c_{{1}}$ $-2\,f_{{1}}{a_{{0}}}^{2}a_{{3}}c_{{0}}$ $-2\,f_{{3}}{a_{{0}}}^{2}a_{{2
}}c_{{3}}$ $-4\,f_{{3}}{a_{{0}}}^{2}a_{{3}}c_{{2}}$ $+2\,f_{{3}}a_{{0}}a_{{1}}a_{{2}}c_{{2}}$ $+
2\,f_{{4}}{a_{{0}}}^{2}a_{{3}}c_{{3}}$ $-2\,f_{{4}}a_{{0}}a_{{1}}a_{{2}}c_{{3}}$ $-6\,f_{{4}}
a_{{0}}a_{{1}}a_{{3}}c_{{2}}$ $-4\,f_{{4}}a_{{0}}{a_{{3}}}^{2}c_{{0}}$ $+2\,f_{{4}}{a_{{1}}}^
{2}a_{{2}}c_{{2}}$ $-f_{{5}}a_{{0}}{a_{{2}}}^{2}c_{{3}}$ $-4\,f_{{5}}a_{{0}}a_{{2}}a_{{3}}c_{
{2}}$ $+12\,f_{{5}}a_{{0}}{a_{{3}}}^{2}c_{{1}}$ $+2\,f_{{5}}{a_{{1}}}^{2}a_{{3}}c_{{2}}$ $-4\,f_
{{5}}a_{{1}}a_{{2}}a_{{3}}c_{{1}}$ $-4\,f_{{5}}a_{{1}}{a_{{3}}}^{2}c_{{0}}$ $-2\,f_{{6}}a_{{0
}}a_{{2}}a_{{3}}c_{{3}}$ $-4\,f_{{6}}a_{{0}}{a_{{3}}}^{2}c_{{2}}$ $-5\,f_{{7}}a_{{0}}{a_{{3}}
}^{2}c_{{3}}$ $-2\,f_{{7}}a_{{1}}{a_{{3}}}^{2}c_{{2}}
-12\,f_{{0}}f_{{5}}{a_{{0}}}^{3}a_{{3}}$ $+4\,f_{{0}}f_{{5}}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $+4
\,f_{{0}}f_{{6}}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $+16\,f_{{0}}f_{{7}}{a_{{0}}}^{2}a_{{2}}a_{{3}
}$ $-4\,f_{{0}}f_{{7}}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $+{f_{{1}}}^{2}{a_{{0}}}^{4}$ $-2\,f_{{1}}f_{
{3}}{a_{{0}}}^{3}a_{{2}}$ $+2\,f_{{1}}f_{{4}}{a_{{0}}}^{3}a_{{3}}$ $-2\,f_{{1}}f_{{4}}{a_{{0}
}}^{2}a_{{1}}a_{{2}}$ $+2\,f_{{1}}f_{{5}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $-2\,f_{{1}}f_{{5}}{a_{
{0}}}^{2}{a_{{2}}}^{2}$ $+8\,f_{{1}}f_{{7}}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $-4\,f_{{1}}f_{{7}}a_{
{0}}a_{{1}}a_{{2}}a_{{3}}$ $+{f_{{3}}}^{2}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $-2\,f_{{3}}f_{{4}}{a_{
{0}}}^{2}a_{{2}}a_{{3}}$ $+2\,f_{{3}}f_{{4}}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $-12\,f_{{3}}f_{{5}}
{a_{{0}}}^{2}{a_{{3}}}^{2}$ $+6\,f_{{3}}f_{{5}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $+2\,f_{{3}}f_{{
7}}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $+{f_{{4}}}^{2}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $-2\,{f_{{4}}}^{2}
a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $+{f_{{4}}}^{2}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $-14\,f_{{4}}f_{{5}}
a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $+6\,f_{{4}}f_{{5}}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $+4\,f_{{4}}f_{{
6}}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $+14\,f_{{4}}f_{{7}}a_{{0}}{a_{{3}}}^{3}$ $-2\,f_{{4}}f_{{7}}
a_{{1}}a_{{2}}{a_{{3}}}^{2}$ $-12\,{f_{{5}}}^{2}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $+{f_{{5}}}^{2}{
a_{{1}}}^{2}{a_{{3}}}^{2}$ $+4\,{f_{{5}}}^{2}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $-12\,f_{{5}}f_{{6}
}a_{{0}}{a_{{3}}}^{3}$ $+8\,f_{{5}}f_{{6}}a_{{1}}a_{{2}}{a_{{3}}}^{2}$ $+6\,f_{{5}}f_{{7}}a_{
{1}}{a_{{3}}}^{3}$ $+{f_{{7}}}^{2}{a_{{3}}}^{4}$.

\noindent
${\bf eq[14]} =
18\,a_{{0}}a_{{1}}c_{{2}}c_{{3}}$ $-16\,a_{{0}}a_{{2}}c_{{1}}c_{{3}}$ $-50\,a_{{0}}a_{{3}}c_{
{1}}c_{{2}}$ $+4\,{a_{{1}}}^{2}c_{{1}}c_{{3}}$ $-14\,{a_{{1}}}^{2}{c_{{2}}}^{2}$ $+2\,a_{{1}}a_{
{2}}c_{{0}}c_{{3}}$ $-4\,{a_{{2}}}^{2}c_{{0}}c_{{2}}$ $+3\,{a_{{2}}}^{2}{c_{{1}}}^{2}$ $+{a_{{3}
}}^{2}{c_{{0}}}^{2}
+
44\,f_{{0}}{a_{{0}}}^{3}c_{{2}}$ $-8\,f_{{0}}{a_{{0}}}^{2}a_{{1}}c_{{1}}$ $+28\,f_{{0}}{a_{{0
}}}^{2}a_{{2}}c_{{0}}$ $+8\,f_{{0}}a_{{0}}{a_{{1}}}^{2}c_{{0}}$ $-4\,f_{{1}}{a_{{0}}}^{2}a_{{
1}}c_{{2}}$ $-24\,f_{{1}}{a_{{0}}}^{2}a_{{2}}c_{{1}}$ $+10\,f_{{1}}{a_{{0}}}^{2}a_{{3}}c_{{0}
}$ $-2\,f_{{1}}a_{{0}}{a_{{1}}}^{2}c_{{1}}$ $+6\,f_{{1}}a_{{0}}a_{{1}}a_{{2}}c_{{0}}$ $-2\,f_{{1
}}{a_{{1}}}^{3}c_{{0}}$ $-6\,f_{{2}}{a_{{0}}}^{2}a_{{1}}c_{{3}}$ $-4\,f_{{2}}{a_{{0}}}^{2}a_{
{2}}c_{{2}}$ $+6\,f_{{2}}{a_{{0}}}^{2}a_{{3}}c_{{1}}$ $+2\,f_{{2}}a_{{0}}{a_{{1}}}^{2}c_{{2}}
-4\,f_{{2}}a_{{0}}a_{{1}}a_{{3}}c_{{0}}$ $+4\,f_{{2}}a_{{0}}{a_{{2}}}^{2}c_{{0}}$ $-2\,f_{{2}
}{a_{{1}}}^{2}a_{{2}}c_{{0}}$ $+18\,f_{{3}}{a_{{0}}}^{2}a_{{2}}c_{{3}}$ $+44\,f_{{3}}{a_{{0}}
}^{2}a_{{3}}c_{{2}}$ $-4\,f_{{3}}a_{{0}}{a_{{1}}}^{2}c_{{3}}$ $-18\,f_{{3}}a_{{0}}a_{{1}}a_{{
2}}c_{{2}}$ $-2\,f_{{3}}a_{{0}}a_{{1}}a_{{3}}c_{{1}}$ $+6\,f_{{3}}a_{{0}}{a_{{2}}}^{2}c_{{1}}
+2\,f_{{3}}{a_{{1}}}^{2}a_{{3}}c_{{0}}$ $-2\,f_{{3}}a_{{1}}{a_{{2}}}^{2}c_{{0}}$ $-30\,f_{{4}
}{a_{{0}}}^{2}a_{{3}}c_{{3}}$ $+30\,f_{{4}}a_{{0}}a_{{1}}a_{{2}}c_{{3}}$ $+70\,f_{{4}}a_{{0}}
a_{{1}}a_{{3}}c_{{2}}$ $+4\,f_{{4}}a_{{0}}{a_{{2}}}^{2}c_{{2}}$ $+40\,f_{{4}}a_{{0}}{a_{{3}}}
^{2}c_{{0}}$ $-4\,f_{{4}}{a_{{1}}}^{3}c_{{3}}$ $-26\,f_{{4}}{a_{{1}}}^{2}a_{{2}}c_{{2}}$ $-4\,f_
{{4}}{a_{{1}}}^{2}a_{{3}}c_{{1}}$ $+4\,f_{{4}}a_{{1}}a_{{2}}a_{{3}}c_{{0}}$ $-4\,f_{{4}}{a_{{
2}}}^{3}c_{{0}}$ $+18\,f_{{5}}a_{{0}}{a_{{2}}}^{2}c_{{3}}$ $+52\,f_{{5}}a_{{0}}a_{{2}}a_{{3}}
c_{{2}}$ $-144\,f_{{5}}a_{{0}}{a_{{3}}}^{2}c_{{1}}$ $-6\,f_{{5}}{a_{{1}}}^{2}a_{{2}}c_{{3}}$ $-
26\,f_{{5}}{a_{{1}}}^{2}a_{{3}}c_{{2}}$ $+54\,f_{{5}}a_{{1}}a_{{2}}a_{{3}}c_{{1}}$ $+54\,f_{{
5}}a_{{1}}{a_{{3}}}^{2}c_{{0}}$ $-6\,f_{{5}}{a_{{2}}}^{3}c_{{1}}$ $-12\,f_{{5}}{a_{{2}}}^{2}a
_{{3}}c_{{0}}$ $+12\,f_{{6}}a_{{0}}a_{{2}}a_{{3}}c_{{3}}$ $+44\,f_{{6}}a_{{0}}{a_{{3}}}^{2}c_
{{2}}$ $-4\,f_{{6}}a_{{1}}{a_{{2}}}^{2}c_{{3}}$ $+4\,f_{{6}}a_{{1}}{a_{{3}}}^{2}c_{{1}}$ $+4\,f_
{{6}}{a_{{2}}}^{3}c_{{2}}$ $-8\,f_{{6}}{a_{{2}}}^{2}a_{{3}}c_{{1}}$ $-8\,f_{{6}}a_{{2}}{a_{{3
}}}^{2}c_{{0}}$ $+4\,f_{{7}}a_{{0}}{a_{{3}}}^{2}c_{{3}}$ $+30\,f_{{7}}a_{{1}}{a_{{3}}}^{2}c_{
{2}}$ $-2\,f_{{7}}{a_{{2}}}^{3}c_{{3}}$ $+14\,f_{{7}}{a_{{2}}}^{2}a_{{3}}c_{{2}}$ $+10\,f_{{7}}a
_{{2}}{a_{{3}}}^{2}c_{{1}}
+
132\,f_{{0}}f_{{5}}{a_{{0}}}^{3}a_{{3}}$ $-28\,f_{{0}}f_{{5}}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $-8
\,f_{{0}}f_{{5}}a_{{0}}{a_{{1}}}^{3}$ $-12\,f_{{0}}f_{{6}}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $-16\,f
_{{0}}f_{{6}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $-128\,f_{{0}}f_{{7}}{a_{{0}}}^{2}a_{{2}}a_{{3}}
-32\,f_{{0}}f_{{7}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $+52\,f_{{0}}f_{{7}}a_{{0}}a_{{1}}{a_{{2}}
}^{2}$ $-11\,{f_{{1}}}^{2}{a_{{0}}}^{4}$ $+4\,f_{{1}}f_{{2}}{a_{{0}}}^{3}a_{{1}}$ $+22\,f_{{1}}f
_{{3}}{a_{{0}}}^{3}a_{{2}}$ $+4\,f_{{1}}f_{{3}}{a_{{0}}}^{2}{a_{{1}}}^{2}$ $-22\,f_{{1}}f_{{4
}}{a_{{0}}}^{3}a_{{3}}$ $+18\,f_{{1}}f_{{4}}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $+4\,f_{{1}}f_{{4}}a
_{{0}}{a_{{1}}}^{3}$ $-30\,f_{{1}}f_{{5}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $+26\,f_{{1}}f_{{5}}{a_
{{0}}}^{2}{a_{{2}}}^{2}$ $-2\,f_{{1}}f_{{5}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $+2\,f_{{1}}f_{{5}}{
a_{{1}}}^{4}$ $+24\,f_{{1}}f_{{6}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $-4\,f_{{1}}f_{{6}}a_{{0}}{a_{
{1}}}^{2}a_{{3}}$ $-8\,f_{{1}}f_{{6}}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $+4\,f_{{1}}f_{{6}}{a_{{1}}
}^{3}a_{{2}}$ $-52\,f_{{1}}f_{{7}}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $+32\,f_{{1}}f_{{7}}a_{{0}}a_{{
1}}a_{{2}}a_{{3}}$ $-2\,f_{{1}}f_{{7}}a_{{0}}{a_{{2}}}^{3}$ $+4\,f_{{1}}f_{{7}}{a_{{1}}}^{3}a
_{{3}}$ $+2\,f_{{1}}f_{{7}}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $+4\,{f_{{2}}}^{2}{a_{{0}}}^{3}a_{{2}}
+4\,f_{{2}}f_{{3}}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $+4\,f_{{2}}f_{{4}}{a_{{0}}}^{2}a_{{1}}a_{{
3}}$ $-8\,f_{{2}}f_{{4}}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $+4\,f_{{2}}f_{{4}}a_{{0}}{a_{{1}}}^{2}a_
{{2}}$ $-16\,f_{{2}}f_{{5}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $+10\,f_{{2}}f_{{5}}a_{{0}}{a_{{1}}}^
{2}a_{{3}}$ $-2\,f_{{2}}f_{{5}}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $+2\,f_{{2}}f_{{5}}{a_{{1}}}^{3}a
_{{2}}$ $+4\,f_{{2}}f_{{6}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $-4\,f_{{2}}f_{{6}}a_{{0}}{a_{{2}}}^
{3}$ $+4\,f_{{2}}f_{{6}}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $+22\,f_{{2}}f_{{7}}a_{{0}}a_{{1}}{a_{{3}
}}^{2}$ $-18\,f_{{2}}f_{{7}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $+4\,f_{{2}}f_{{7}}{a_{{1}}}^{2}a_{{
2}}a_{{3}}$ $+2\,f_{{2}}f_{{7}}a_{{1}}{a_{{2}}}^{3}$ $-11\,{f_{{3}}}^{2}{a_{{0}}}^{2}{a_{{2}}
}^{2}$ $+22\,f_{{3}}f_{{4}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $+4\,f_{{3}}f_{{4}}a_{{0}}{a_{{1}}}^{
2}a_{{3}}$ $-26\,f_{{3}}f_{{4}}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $+132\,f_{{3}}f_{{5}}{a_{{0}}}^{2
}{a_{{3}}}^{2}$ $-50\,f_{{3}}f_{{5}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $-8\,f_{{3}}f_{{5}}a_{{0}}{
a_{{2}}}^{3}$ $-2\,f_{{3}}f_{{5}}{a_{{1}}}^{3}a_{{3}}$ $+2\,f_{{3}}f_{{5}}{a_{{1}}}^{2}{a_{{2
}}}^{2}$ $-4\,f_{{3}}f_{{6}}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $+4\,f_{{3}}f_{{6}}a_{{1}}{a_{{2}}}^
{3}$ $-22\,f_{{3}}f_{{7}}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $-4\,f_{{3}}f_{{7}}{a_{{1}}}^{2}{a_{{3}
}}^{2}$ $+2\,f_{{3}}f_{{7}}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $+2\,f_{{3}}f_{{7}}{a_{{2}}}^{4}$ $-11\,
{f_{{4}}}^{2}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $+18\,{f_{{4}}}^{2}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $+4
\,{f_{{4}}}^{2}a_{{0}}{a_{{2}}}^{3}$ $+4\,{f_{{4}}}^{2}{a_{{1}}}^{3}a_{{3}}$ $-15\,{f_{{4}}}^
{2}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $+162\,f_{{4}}f_{{5}}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $+12\,f_{{4}
}f_{{5}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $-74\,f_{{4}}f_{{5}}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $-36\,f
_{{4}}f_{{6}}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $-4\,f_{{4}}f_{{6}}{a_{{1}}}^{2}{a_{{3}}}^{2}$ $+4
\,f_{{4}}f_{{6}}{a_{{2}}}^{4}$ $-142\,f_{{4}}f_{{7}}a_{{0}}{a_{{3}}}^{3}$ $+6\,f_{{4}}f_{{7}}
a_{{1}}a_{{2}}{a_{{3}}}^{2}$ $+16\,f_{{4}}f_{{7}}{a_{{2}}}^{3}a_{{3}}$ $+148\,{f_{{5}}}^{2}a_
{{0}}a_{{2}}{a_{{3}}}^{2}$ $-7\,{f_{{5}}}^{2}{a_{{1}}}^{2}{a_{{3}}}^{2}$ $-54\,{f_{{5}}}^{2}a
_{{1}}{a_{{2}}}^{2}a_{{3}}$ $+3\,{f_{{5}}}^{2}{a_{{2}}}^{4}$ $+132\,f_{{5}}f_{{6}}a_{{0}}{a_{
{3}}}^{3}$ $-108\,f_{{5}}f_{{6}}a_{{1}}a_{{2}}{a_{{3}}}^{2}$ $+20\,f_{{5}}f_{{6}}{a_{{2}}}^{3
}a_{{3}}$ $-80\,f_{{5}}f_{{7}}a_{{1}}{a_{{3}}}^{3}$ $+34\,f_{{5}}f_{{7}}{a_{{2}}}^{2}{a_{{3}}
}^{2}$ $+12\,{f_{{6}}}^{2}{a_{{2}}}^{2}{a_{{3}}}^{2}$ $+12\,f_{{6}}f_{{7}}a_{{2}}{a_{{3}}}^{3
}$ $-20\,{f_{{7}}}^{2}{a_{{3}}}^{4}$.

\noindent
${\bf eq[15]} =
-9\,a_{{0}}a_{{1}}c_{{2}}c_{{3}}$ $+8\,a_{{0}}a_{{2}}c_{{1}}c_{{3}}$ $+27\,a_{{0}}a_{{3}}c_{{
1}}c_{{2}}$ $-2\,{a_{{1}}}^{2}c_{{1}}c_{{3}}$ $+7\,{a_{{1}}}^{2}{c_{{2}}}^{2}$ $-a_{{1}}a_{{2}}c
_{{0}}c_{{3}}$ $+2\,{a_{{2}}}^{2}c_{{0}}c_{{2}}$ $-{a_{{2}}}^{2}{c_{{1}}}^{2}$ $+a_{{2}}a_{{3}}c
_{{0}}c_{{1}}
-24\,f_{{0}}{a_{{0}}}^{3}c_{{2}}$ $+4\,f_{{0}}{a_{{0}}}^{2}a_{{1}}c_{{1}}$ $-16\,f_{{0}}{a_{{0
}}}^{2}a_{{2}}c_{{0}}$ $-4\,f_{{0}}a_{{0}}{a_{{1}}}^{2}c_{{0}}$ $+2\,f_{{1}}{a_{{0}}}^{2}a_{{
1}}c_{{2}}$ $+13\,f_{{1}}{a_{{0}}}^{2}a_{{2}}c_{{1}}$ $-6\,f_{{1}}{a_{{0}}}^{2}a_{{3}}c_{{0}}
+f_{{1}}a_{{0}}{a_{{1}}}^{2}c_{{1}}$ $-3\,f_{{1}}a_{{0}}a_{{1}}a_{{2}}c_{{0}}$ $+f_{{1}}{a_{{
1}}}^{3}c_{{0}}$ $+3\,f_{{2}}{a_{{0}}}^{2}a_{{1}}c_{{3}}$ $+2\,f_{{2}}{a_{{0}}}^{2}a_{{2}}c_{
{2}}$ $-3\,f_{{2}}{a_{{0}}}^{2}a_{{3}}c_{{1}}$ $-f_{{2}}a_{{0}}{a_{{1}}}^{2}c_{{2}}$ $+2\,f_{{2}
}a_{{0}}a_{{1}}a_{{3}}c_{{0}}$ $-2\,f_{{2}}a_{{0}}{a_{{2}}}^{2}c_{{0}}$ $+f_{{2}}{a_{{1}}}^{2
}a_{{2}}c_{{0}}$ $-10\,f_{{3}}{a_{{0}}}^{2}a_{{2}}c_{{3}}$ $-24\,f_{{3}}{a_{{0}}}^{2}a_{{3}}c
_{{2}}$ $+2\,f_{{3}}a_{{0}}{a_{{1}}}^{2}c_{{3}}$ $+10\,f_{{3}}a_{{0}}a_{{1}}a_{{2}}c_{{2}}$ $+f_
{{3}}a_{{0}}a_{{1}}a_{{3}}c_{{1}}$ $-3\,f_{{3}}a_{{0}}{a_{{2}}}^{2}c_{{1}}$ $-f_{{3}}{a_{{1}}
}^{2}a_{{3}}c_{{0}}$ $+f_{{3}}a_{{1}}{a_{{2}}}^{2}c_{{0}}$ $+16\,f_{{4}}{a_{{0}}}^{2}a_{{3}}c
_{{3}}$ $-16\,f_{{4}}a_{{0}}a_{{1}}a_{{2}}c_{{3}}$ $-38\,f_{{4}}a_{{0}}a_{{1}}a_{{3}}c_{{2}}$ $-
2\,f_{{4}}a_{{0}}{a_{{2}}}^{2}c_{{2}}$ $-22\,f_{{4}}a_{{0}}{a_{{3}}}^{2}c_{{0}}$ $+2\,f_{{4}}
{a_{{1}}}^{3}c_{{3}}$ $+14\,f_{{4}}{a_{{1}}}^{2}a_{{2}}c_{{2}}$ $+2\,f_{{4}}{a_{{1}}}^{2}a_{{
3}}c_{{1}}$ $-2\,f_{{4}}a_{{1}}a_{{2}}a_{{3}}c_{{0}}$ $+2\,f_{{4}}{a_{{2}}}^{3}c_{{0}}$ $-9\,f_{
{5}}a_{{0}}{a_{{2}}}^{2}c_{{3}}$ $-28\,f_{{5}}a_{{0}}a_{{2}}a_{{3}}c_{{2}}$ $+78\,f_{{5}}a_{{0
}}{a_{{3}}}^{2}c_{{1}}$ $+3\,f_{{5}}{a_{{1}}}^{2}a_{{2}}c_{{3}}$ $+13\,f_{{5}}{a_{{1}}}^{2}a_
{{3}}c_{{2}}$ $-28\,f_{{5}}a_{{1}}a_{{2}}a_{{3}}c_{{1}}$ $-28\,f_{{5}}a_{{1}}{a_{{3}}}^{2}c_{
{0}}$ $+2\,f_{{5}}{a_{{2}}}^{3}c_{{1}}$ $+5\,f_{{5}}{a_{{2}}}^{2}a_{{3}}c_{{0}}$ $-6\,f_{{6}}a_{
{0}}a_{{2}}a_{{3}}c_{{3}}$ $-24\,f_{{6}}a_{{0}}{a_{{3}}}^{2}c_{{2}}$ $+2\,f_{{6}}a_{{1}}{a_{{
2}}}^{2}c_{{3}}$ $-2\,f_{{6}}a_{{1}}{a_{{3}}}^{2}c_{{1}}$ $-2\,f_{{6}}{a_{{2}}}^{3}c_{{2}}$ $+2
\,f_{{6}}{a_{{2}}}^{2}a_{{3}}c_{{1}}$ $+2\,f_{{6}}a_{{2}}{a_{{3}}}^{2}c_{{0}}$ $-6\,f_{{7}}a_
{{0}}{a_{{3}}}^{2}c_{{3}}$ $-13\,f_{{7}}a_{{1}}{a_{{3}}}^{2}c_{{2}}$ $+f_{{7}}{a_{{2}}}^{3}c_
{{3}}$ $-7\,f_{{7}}{a_{{2}}}^{2}a_{{3}}c_{{2}}$ $-5\,f_{{7}}a_{{2}}{a_{{3}}}^{2}c_{{1}}
-72\,f_{{0}}f_{{5}}{a_{{0}}}^{3}a_{{3}}$ $+16\,f_{{0}}f_{{5}}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $+4
\,f_{{0}}f_{{5}}a_{{0}}{a_{{1}}}^{3}$ $+8\,f_{{0}}f_{{6}}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $+8\,f_{
{0}}f_{{6}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $+72\,f_{{0}}f_{{7}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $+16
\,f_{{0}}f_{{7}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $-28\,f_{{0}}f_{{7}}a_{{0}}a_{{1}}{a_{{2}}}^{
2}$ $+6\,{f_{{1}}}^{2}{a_{{0}}}^{4}$ $-2\,f_{{1}}f_{{2}}{a_{{0}}}^{3}a_{{1}}$ $-12\,f_{{1}}f_{{3
}}{a_{{0}}}^{3}a_{{2}}$ $-2\,f_{{1}}f_{{3}}{a_{{0}}}^{2}{a_{{1}}}^{2}$ $+12\,f_{{1}}f_{{4}}{a
_{{0}}}^{3}a_{{3}}$ $-10\,f_{{1}}f_{{4}}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $-2\,f_{{1}}f_{{4}}a_{{0
}}{a_{{1}}}^{3}$ $+16\,f_{{1}}f_{{5}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $-14\,f_{{1}}f_{{5}}{a_{{0}
}}^{2}{a_{{2}}}^{2}$ $+f_{{1}}f_{{5}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $-f_{{1}}f_{{5}}{a_{{1}}}^{
4}$ $-12\,f_{{1}}f_{{6}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $+2\,f_{{1}}f_{{6}}a_{{0}}{a_{{1}}}^{2}a
_{{3}}$ $+4\,f_{{1}}f_{{6}}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $-2\,f_{{1}}f_{{6}}{a_{{1}}}^{3}a_{{2
}}$ $+30\,f_{{1}}f_{{7}}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $-18\,f_{{1}}f_{{7}}a_{{0}}a_{{1}}a_{{2}}
a_{{3}}$ $+f_{{1}}f_{{7}}a_{{0}}{a_{{2}}}^{3}$ $-2\,f_{{1}}f_{{7}}{a_{{1}}}^{3}a_{{3}}$ $-f_{{1}
}f_{{7}}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $-2\,{f_{{2}}}^{2}{a_{{0}}}^{3}a_{{2}}$ $-2\,f_{{2}}f_{{3
}}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $-2\,f_{{2}}f_{{4}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $+4\,f_{{2}}f_
{{4}}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $-2\,f_{{2}}f_{{4}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $+8\,f_{{2}}
f_{{5}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $-5\,f_{{2}}f_{{5}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $+f_{{2}}
f_{{5}}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $-f_{{2}}f_{{5}}{a_{{1}}}^{3}a_{{2}}$ $-2\,f_{{2}}f_{{6}}
a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $+2\,f_{{2}}f_{{6}}a_{{0}}{a_{{2}}}^{3}$ $-2\,f_{{2}}f_{{6}}{a_
{{1}}}^{2}{a_{{2}}}^{2}$ $-11\,f_{{2}}f_{{7}}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $+9\,f_{{2}}f_{{7}}
a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $-2\,f_{{2}}f_{{7}}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $-f_{{2}}f_{{7}}
a_{{1}}{a_{{2}}}^{3}$ $+6\,{f_{{3}}}^{2}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $-12\,f_{{3}}f_{{4}}{a_{{0
}}}^{2}a_{{2}}a_{{3}}$ $-2\,f_{{3}}f_{{4}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $+14\,f_{{3}}f_{{4}}a_
{{0}}a_{{1}}{a_{{2}}}^{2}$ $-72\,f_{{3}}f_{{5}}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $+28\,f_{{3}}f_{{5
}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $+4\,f_{{3}}f_{{5}}a_{{0}}{a_{{2}}}^{3}$ $+f_{{3}}f_{{5}}{a_{
{1}}}^{3}a_{{3}}$ $-f_{{3}}f_{{5}}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $+2\,f_{{3}}f_{{6}}{a_{{1}}}^{2
}a_{{2}}a_{{3}}$ $-2\,f_{{3}}f_{{6}}a_{{1}}{a_{{2}}}^{3}$ $+12\,f_{{3}}f_{{7}}a_{{0}}a_{{2}}{
a_{{3}}}^{2}$ $+2\,f_{{3}}f_{{7}}{a_{{1}}}^{2}{a_{{3}}}^{2}$ $-f_{{3}}f_{{7}}a_{{1}}{a_{{2}}}
^{2}a_{{3}}$ $-f_{{3}}f_{{7}}{a_{{2}}}^{4}$ $+6\,{f_{{4}}}^{2}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $-10\,
{f_{{4}}}^{2}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $-2\,{f_{{4}}}^{2}a_{{0}}{a_{{2}}}^{3}$ $-2\,{f_{{
4}}}^{2}{a_{{1}}}^{3}a_{{3}}$ $+8\,{f_{{4}}}^{2}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $-88\,f_{{4}}f_{{
5}}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $-6\,f_{{4}}f_{{5}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $+40\,f_{{4}}
f_{{5}}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $+20\,f_{{4}}f_{{6}}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $+2\,f_{
{4}}f_{{6}}{a_{{1}}}^{2}{a_{{3}}}^{2}$ $-2\,f_{{4}}f_{{6}}{a_{{2}}}^{4}$ $+78\,f_{{4}}f_{{7}}
a_{{0}}{a_{{3}}}^{3}$ $-4\,f_{{4}}f_{{7}}a_{{1}}a_{{2}}{a_{{3}}}^{2}$ $-8\,f_{{4}}f_{{7}}{a_{
{2}}}^{3}a_{{3}}$ $-80\,{f_{{5}}}^{2}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $+4\,{f_{{5}}}^{2}{a_{{1}}}
^{2}{a_{{3}}}^{2}$ $+28\,{f_{{5}}}^{2}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $-{f_{{5}}}^{2}{a_{{2}}}^{
4}$ $-72\,f_{{5}}f_{{6}}a_{{0}}{a_{{3}}}^{3}$ $+56\,f_{{5}}f_{{6}}a_{{1}}a_{{2}}{a_{{3}}}^{2}
-8\,f_{{5}}f_{{6}}{a_{{2}}}^{3}a_{{3}}$ $+46\,f_{{5}}f_{{7}}a_{{1}}{a_{{3}}}^{3}$ $-17\,f_{{5
}}f_{{7}}{a_{{2}}}^{2}{a_{{3}}}^{2}$ $-4\,{f_{{6}}}^{2}{a_{{2}}}^{2}{a_{{3}}}^{2}$ $-6\,f_{{6
}}f_{{7}}a_{{2}}{a_{{3}}}^{3}$ $+6\,{f_{{7}}}^{2}{a_{{3}}}^{4}$.

\noindent
${\bf eq[16]} =
9\,a_{{0}}a_{{1}}c_{{2}}c_{{3}}$ $-9\,a_{{0}}a_{{2}}c_{{1}}c_{{3}}$ $-27\,a_{{0}}a_{{3}}c_{{1
}}c_{{2}}$ $+2\,{a_{{1}}}^{2}c_{{1}}c_{{3}}$ $-7\,{a_{{1}}}^{2}{c_{{2}}}^{2}$ $+a_{{1}}a_{{2}}c_
{{0}}c_{{3}}$ $+a_{{1}}a_{{2}}c_{{1}}c_{{2}}$ $-2\,{a_{{2}}}^{2}c_{{0}}c_{{2}}$ $+2\,{a_{{2}}}^{
2}{c_{{1}}}^{2}
+
24\,f_{{0}}{a_{{0}}}^{3}c_{{2}}$ $-4\,f_{{0}}{a_{{0}}}^{2}a_{{1}}c_{{1}}$ $+16\,f_{{0}}{a_{{0
}}}^{2}a_{{2}}c_{{0}}$ $+4\,f_{{0}}a_{{0}}{a_{{1}}}^{2}c_{{0}}$ $-2\,f_{{1}}{a_{{0}}}^{2}a_{{
1}}c_{{2}}$ $-13\,f_{{1}}{a_{{0}}}^{2}a_{{2}}c_{{1}}$ $+6\,f_{{1}}{a_{{0}}}^{2}a_{{3}}c_{{0}}
-f_{{1}}a_{{0}}{a_{{1}}}^{2}c_{{1}}$ $+3\,f_{{1}}a_{{0}}a_{{1}}a_{{2}}c_{{0}}$ $-f_{{1}}{a_{{
1}}}^{3}c_{{0}}$ $-3\,f_{{2}}{a_{{0}}}^{2}a_{{1}}c_{{3}}$ $-2\,f_{{2}}{a_{{0}}}^{2}a_{{2}}c_{
{2}}$ $+3\,f_{{2}}{a_{{0}}}^{2}a_{{3}}c_{{1}}$ $+f_{{2}}a_{{0}}{a_{{1}}}^{2}c_{{2}}$ $-2\,f_{{2}
}a_{{0}}a_{{1}}a_{{3}}c_{{0}}$ $+2\,f_{{2}}a_{{0}}{a_{{2}}}^{2}c_{{0}}$ $-f_{{2}}{a_{{1}}}^{2
}a_{{2}}c_{{0}}$ $+10\,f_{{3}}{a_{{0}}}^{2}a_{{2}}c_{{3}}$ $+24\,f_{{3}}{a_{{0}}}^{2}a_{{3}}c
_{{2}}$ $-2\,f_{{3}}a_{{0}}{a_{{1}}}^{2}c_{{3}}$ $-10\,f_{{3}}a_{{0}}a_{{1}}a_{{2}}c_{{2}}$ $-f_
{{3}}a_{{0}}a_{{1}}a_{{3}}c_{{1}}$ $+3\,f_{{3}}a_{{0}}{a_{{2}}}^{2}c_{{1}}$ $+f_{{3}}{a_{{1}}
}^{2}a_{{3}}c_{{0}}$ $-f_{{3}}a_{{1}}{a_{{2}}}^{2}c_{{0}}$ $-16\,f_{{4}}{a_{{0}}}^{2}a_{{3}}c
_{{3}}$ $+16\,f_{{4}}a_{{0}}a_{{1}}a_{{2}}c_{{3}}$ $+38\,f_{{4}}a_{{0}}a_{{1}}a_{{3}}c_{{2}}$ $+
2\,f_{{4}}a_{{0}}{a_{{2}}}^{2}c_{{2}}$ $+22\,f_{{4}}a_{{0}}{a_{{3}}}^{2}c_{{0}}$ $-2\,f_{{4}}
{a_{{1}}}^{3}c_{{3}}$ $-14\,f_{{4}}{a_{{1}}}^{2}a_{{2}}c_{{2}}$ $-2\,f_{{4}}{a_{{1}}}^{2}a_{{
3}}c_{{1}}$ $+2\,f_{{4}}a_{{1}}a_{{2}}a_{{3}}c_{{0}}$ $-2\,f_{{4}}{a_{{2}}}^{3}c_{{0}}$ $+9\,f_{
{5}}a_{{0}}{a_{{2}}}^{2}c_{{3}}$ $+28\,f_{{5}}a_{{0}}a_{{2}}a_{{3}}c_{{2}}$ $-78\,f_{{5}}a_{{0
}}{a_{{3}}}^{2}c_{{1}}$ $-3\,f_{{5}}{a_{{1}}}^{2}a_{{2}}c_{{3}}$ $-13\,f_{{5}}{a_{{1}}}^{2}a_
{{3}}c_{{2}}$ $+30\,f_{{5}}a_{{1}}a_{{2}}a_{{3}}c_{{1}}$ $+28\,f_{{5}}a_{{1}}{a_{{3}}}^{2}c_{
{0}}$ $-3\,f_{{5}}{a_{{2}}}^{3}c_{{1}}$ $-5\,f_{{5}}{a_{{2}}}^{2}a_{{3}}c_{{0}}$ $+6\,f_{{6}}a_{
{0}}a_{{2}}a_{{3}}c_{{3}}$ $+24\,f_{{6}}a_{{0}}{a_{{3}}}^{2}c_{{2}}$ $-2\,f_{{6}}a_{{1}}{a_{{
2}}}^{2}c_{{3}}$ $+2\,f_{{6}}a_{{1}}{a_{{3}}}^{2}c_{{1}}$ $+2\,f_{{6}}{a_{{2}}}^{3}c_{{2}}$ $-4
\,f_{{6}}{a_{{2}}}^{2}a_{{3}}c_{{1}}$ $-2\,f_{{6}}a_{{2}}{a_{{3}}}^{2}c_{{0}}$ $+6\,f_{{7}}a_
{{0}}{a_{{3}}}^{2}c_{{3}}$ $+13\,f_{{7}}a_{{1}}{a_{{3}}}^{2}c_{{2}}$ $-f_{{7}}{a_{{2}}}^{3}c_
{{3}}$ $+7\,f_{{7}}{a_{{2}}}^{2}a_{{3}}c_{{2}}$ $+2\,f_{{7}}a_{{2}}{a_{{3}}}^{2}c_{{1}}
+
72\,f_{{0}}f_{{5}}{a_{{0}}}^{3}a_{{3}}$ $-16\,f_{{0}}f_{{5}}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $-4
\,f_{{0}}f_{{5}}a_{{0}}{a_{{1}}}^{3}$ $-8\,f_{{0}}f_{{6}}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $-8\,f_{
{0}}f_{{6}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $-72\,f_{{0}}f_{{7}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $-16
\,f_{{0}}f_{{7}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $+28\,f_{{0}}f_{{7}}a_{{0}}a_{{1}}{a_{{2}}}^{
2}$ $-6\,{f_{{1}}}^{2}{a_{{0}}}^{4}$ $+2\,f_{{1}}f_{{2}}{a_{{0}}}^{3}a_{{1}}$ $+12\,f_{{1}}f_{{3
}}{a_{{0}}}^{3}a_{{2}}$ $+2\,f_{{1}}f_{{3}}{a_{{0}}}^{2}{a_{{1}}}^{2}$ $-12\,f_{{1}}f_{{4}}{a
_{{0}}}^{3}a_{{3}}$ $+10\,f_{{1}}f_{{4}}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $+2\,f_{{1}}f_{{4}}a_{{0
}}{a_{{1}}}^{3}$ $-16\,f_{{1}}f_{{5}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $+14\,f_{{1}}f_{{5}}{a_{{0}
}}^{2}{a_{{2}}}^{2}$ $-f_{{1}}f_{{5}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $+f_{{1}}f_{{5}}{a_{{1}}}^{
4}$ $+12\,f_{{1}}f_{{6}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $-2\,f_{{1}}f_{{6}}a_{{0}}{a_{{1}}}^{2}a
_{{3}}$ $-4\,f_{{1}}f_{{6}}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $+2\,f_{{1}}f_{{6}}{a_{{1}}}^{3}a_{{2
}}$ $-30\,f_{{1}}f_{{7}}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $+18\,f_{{1}}f_{{7}}a_{{0}}a_{{1}}a_{{2}}
a_{{3}}$ $-f_{{1}}f_{{7}}a_{{0}}{a_{{2}}}^{3}$ $+2\,f_{{1}}f_{{7}}{a_{{1}}}^{3}a_{{3}}$ $+f_{{1}
}f_{{7}}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $+2\,{f_{{2}}}^{2}{a_{{0}}}^{3}a_{{2}}$ $+2\,f_{{2}}f_{{3
}}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $+2\,f_{{2}}f_{{4}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $-4\,f_{{2}}f_
{{4}}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $+2\,f_{{2}}f_{{4}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $-8\,f_{{2}}
f_{{5}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $+5\,f_{{2}}f_{{5}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $-f_{{2}}
f_{{5}}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $+f_{{2}}f_{{5}}{a_{{1}}}^{3}a_{{2}}$ $+2\,f_{{2}}f_{{6}}
a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $-2\,f_{{2}}f_{{6}}a_{{0}}{a_{{2}}}^{3}$ $+2\,f_{{2}}f_{{6}}{a_
{{1}}}^{2}{a_{{2}}}^{2}$ $+11\,f_{{2}}f_{{7}}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $-9\,f_{{2}}f_{{7}}
a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $+2\,f_{{2}}f_{{7}}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $+f_{{2}}f_{{7}}
a_{{1}}{a_{{2}}}^{3}$ $-6\,{f_{{3}}}^{2}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $+12\,f_{{3}}f_{{4}}{a_{{0
}}}^{2}a_{{2}}a_{{3}}$ $+2\,f_{{3}}f_{{4}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $-14\,f_{{3}}f_{{4}}a_
{{0}}a_{{1}}{a_{{2}}}^{2}$ $+72\,f_{{3}}f_{{5}}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $-28\,f_{{3}}f_{{5
}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $-4\,f_{{3}}f_{{5}}a_{{0}}{a_{{2}}}^{3}$ $-f_{{3}}f_{{5}}{a_{
{1}}}^{3}a_{{3}}$ $+f_{{3}}f_{{5}}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $-2\,f_{{3}}f_{{6}}{a_{{1}}}^{2
}a_{{2}}a_{{3}}$ $+2\,f_{{3}}f_{{6}}a_{{1}}{a_{{2}}}^{3}$ $-12\,f_{{3}}f_{{7}}a_{{0}}a_{{2}}{
a_{{3}}}^{2}$ $-2\,f_{{3}}f_{{7}}{a_{{1}}}^{2}{a_{{3}}}^{2}$ $+f_{{3}}f_{{7}}a_{{1}}{a_{{2}}}
^{2}a_{{3}}$ $+f_{{3}}f_{{7}}{a_{{2}}}^{4}$ $-6\,{f_{{4}}}^{2}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $+10\,
{f_{{4}}}^{2}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $+2\,{f_{{4}}}^{2}a_{{0}}{a_{{2}}}^{3}$ $+2\,{f_{{
4}}}^{2}{a_{{1}}}^{3}a_{{3}}$ $-8\,{f_{{4}}}^{2}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $+88\,f_{{4}}f_{{
5}}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $+6\,f_{{4}}f_{{5}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $-40\,f_{{4}}
f_{{5}}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $-20\,f_{{4}}f_{{6}}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $-2\,f_{
{4}}f_{{6}}{a_{{1}}}^{2}{a_{{3}}}^{2}$ $+2\,f_{{4}}f_{{6}}{a_{{2}}}^{4}$ $-78\,f_{{4}}f_{{7}}
a_{{0}}{a_{{3}}}^{3}$ $+4\,f_{{4}}f_{{7}}a_{{1}}a_{{2}}{a_{{3}}}^{2}$ $+8\,f_{{4}}f_{{7}}{a_{
{2}}}^{3}a_{{3}}$ $+80\,{f_{{5}}}^{2}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $-4\,{f_{{5}}}^{2}{a_{{1}}}
^{2}{a_{{3}}}^{2}$ $-28\,{f_{{5}}}^{2}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $+{f_{{5}}}^{2}{a_{{2}}}^{
4}$ $+72\,f_{{5}}f_{{6}}a_{{0}}{a_{{3}}}^{3}$ $-56\,f_{{5}}f_{{6}}a_{{1}}a_{{2}}{a_{{3}}}^{2}
+8\,f_{{5}}f_{{6}}{a_{{2}}}^{3}a_{{3}}$ $-46\,f_{{5}}f_{{7}}a_{{1}}{a_{{3}}}^{3}$ $+17\,f_{{5
}}f_{{7}}{a_{{2}}}^{2}{a_{{3}}}^{2}$ $+4\,{f_{{6}}}^{2}{a_{{2}}}^{2}{a_{{3}}}^{2}$ $+6\,f_{{6
}}f_{{7}}a_{{2}}{a_{{3}}}^{3}$ $-6\,{f_{{7}}}^{2}{a_{{3}}}^{4}$.

\noindent
${\bf eq[17]} =
-10\,a_{{0}}a_{{1}}c_{{2}}c_{{3}}$ $+9\,a_{{0}}a_{{2}}c_{{1}}c_{{3}}$ $+27\,a_{{0}}a_{{3}}c_{
{1}}c_{{2}}$ $-2\,{a_{{1}}}^{2}c_{{1}}c_{{3}}$ $+8\,{a_{{1}}}^{2}{c_{{2}}}^{2}$ $-a_{{1}}a_{{2}}
c_{{0}}c_{{3}}$ $+a_{{1}}a_{{3}}c_{{0}}c_{{2}}$ $+2\,{a_{{2}}}^{2}c_{{0}}c_{{2}}$ $-2\,{a_{{2}}}
^{2}{c_{{1}}}^{2}
-24\,f_{{0}}{a_{{0}}}^{3}c_{{2}}$ $+4\,f_{{0}}{a_{{0}}}^{2}a_{{1}}c_{{1}}$ $-16\,f_{{0}}{a_{{0
}}}^{2}a_{{2}}c_{{0}}$ $-4\,f_{{0}}a_{{0}}{a_{{1}}}^{2}c_{{0}}$ $+2\,f_{{1}}{a_{{0}}}^{2}a_{{
1}}c_{{2}}$ $+13\,f_{{1}}{a_{{0}}}^{2}a_{{2}}c_{{1}}$ $-6\,f_{{1}}{a_{{0}}}^{2}a_{{3}}c_{{0}}
+f_{{1}}a_{{0}}{a_{{1}}}^{2}c_{{1}}$ $-3\,f_{{1}}a_{{0}}a_{{1}}a_{{2}}c_{{0}}$ $+f_{{1}}{a_{{
1}}}^{3}c_{{0}}$ $+3\,f_{{2}}{a_{{0}}}^{2}a_{{1}}c_{{3}}$ $+2\,f_{{2}}{a_{{0}}}^{2}a_{{2}}c_{
{2}}$ $-3\,f_{{2}}{a_{{0}}}^{2}a_{{3}}c_{{1}}$ $-f_{{2}}a_{{0}}{a_{{1}}}^{2}c_{{2}}$ $+2\,f_{{2}
}a_{{0}}a_{{1}}a_{{3}}c_{{0}}$ $-2\,f_{{2}}a_{{0}}{a_{{2}}}^{2}c_{{0}}$ $+f_{{2}}{a_{{1}}}^{2
}a_{{2}}c_{{0}}$ $-10\,f_{{3}}{a_{{0}}}^{2}a_{{2}}c_{{3}}$ $-24\,f_{{3}}{a_{{0}}}^{2}a_{{3}}c
_{{2}}$ $+2\,f_{{3}}a_{{0}}{a_{{1}}}^{2}c_{{3}}$ $+10\,f_{{3}}a_{{0}}a_{{1}}a_{{2}}c_{{2}}$ $+f_
{{3}}a_{{0}}a_{{1}}a_{{3}}c_{{1}}$ $-3\,f_{{3}}a_{{0}}{a_{{2}}}^{2}c_{{1}}$ $-f_{{3}}{a_{{1}}
}^{2}a_{{3}}c_{{0}}$ $+f_{{3}}a_{{1}}{a_{{2}}}^{2}c_{{0}}$ $+16\,f_{{4}}{a_{{0}}}^{2}a_{{3}}c
_{{3}}$ $-16\,f_{{4}}a_{{0}}a_{{1}}a_{{2}}c_{{3}}$ $-38\,f_{{4}}a_{{0}}a_{{1}}a_{{3}}c_{{2}}$ $-
2\,f_{{4}}a_{{0}}{a_{{2}}}^{2}c_{{2}}$ $-22\,f_{{4}}a_{{0}}{a_{{3}}}^{2}c_{{0}}$ $+2\,f_{{4}}
{a_{{1}}}^{3}c_{{3}}$ $+14\,f_{{4}}{a_{{1}}}^{2}a_{{2}}c_{{2}}$ $+2\,f_{{4}}{a_{{1}}}^{2}a_{{
3}}c_{{1}}$ $-2\,f_{{4}}a_{{1}}a_{{2}}a_{{3}}c_{{0}}$ $+2\,f_{{4}}{a_{{2}}}^{3}c_{{0}}$ $-10\,f_
{{5}}a_{{0}}{a_{{2}}}^{2}c_{{3}}$ $-28\,f_{{5}}a_{{0}}a_{{2}}a_{{3}}c_{{2}}$ $+78\,f_{{5}}a_{
{0}}{a_{{3}}}^{2}c_{{1}}$ $+3\,f_{{5}}{a_{{1}}}^{2}a_{{2}}c_{{3}}$ $+15\,f_{{5}}{a_{{1}}}^{2}
a_{{3}}c_{{2}}$ $-30\,f_{{5}}a_{{1}}a_{{2}}a_{{3}}c_{{1}}$ $-28\,f_{{5}}a_{{1}}{a_{{3}}}^{2}c
_{{0}}$ $+4\,f_{{5}}{a_{{2}}}^{3}c_{{1}}$ $+6\,f_{{5}}{a_{{2}}}^{2}a_{{3}}c_{{0}}$ $-8\,f_{{6}}a
_{{0}}a_{{2}}a_{{3}}c_{{3}}$ $-24\,f_{{6}}a_{{0}}{a_{{3}}}^{2}c_{{2}}$ $+2\,f_{{6}}a_{{1}}{a_
{{2}}}^{2}c_{{3}}$ $-2\,f_{{6}}a_{{1}}{a_{{3}}}^{2}c_{{1}}$ $-2\,f_{{6}}{a_{{2}}}^{3}c_{{2}}$ $+
6\,f_{{6}}{a_{{2}}}^{2}a_{{3}}c_{{1}}$ $+4\,f_{{6}}a_{{2}}{a_{{3}}}^{2}c_{{0}}$ $-6\,f_{{7}}a
_{{0}}{a_{{3}}}^{2}c_{{3}}$ $-16\,f_{{7}}a_{{1}}{a_{{3}}}^{2}c_{{2}}$ $+f_{{7}}{a_{{2}}}^{3}c
_{{3}}$ $-7\,f_{{7}}{a_{{2}}}^{2}a_{{3}}c_{{2}}$ $-2\,f_{{7}}a_{{2}}{a_{{3}}}^{2}c_{{1}}
-72\,f_{{0}}f_{{5}}{a_{{0}}}^{3}a_{{3}}$ $+16\,f_{{0}}f_{{5}}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $+4
\,f_{{0}}f_{{5}}a_{{0}}{a_{{1}}}^{3}$ $+8\,f_{{0}}f_{{6}}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $+8\,f_{
{0}}f_{{6}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $+72\,f_{{0}}f_{{7}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $+16
\,f_{{0}}f_{{7}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $-28\,f_{{0}}f_{{7}}a_{{0}}a_{{1}}{a_{{2}}}^{
2}$ $+6\,{f_{{1}}}^{2}{a_{{0}}}^{4}$ $-2\,f_{{1}}f_{{2}}{a_{{0}}}^{3}a_{{1}}$ $-12\,f_{{1}}f_{{3
}}{a_{{0}}}^{3}a_{{2}}$ $-2\,f_{{1}}f_{{3}}{a_{{0}}}^{2}{a_{{1}}}^{2}$ $+12\,f_{{1}}f_{{4}}{a
_{{0}}}^{3}a_{{3}}$ $-10\,f_{{1}}f_{{4}}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $-2\,f_{{1}}f_{{4}}a_{{0
}}{a_{{1}}}^{3}$ $+16\,f_{{1}}f_{{5}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $-14\,f_{{1}}f_{{5}}{a_{{0}
}}^{2}{a_{{2}}}^{2}$ $+f_{{1}}f_{{5}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $-f_{{1}}f_{{5}}{a_{{1}}}^{
4}$ $-12\,f_{{1}}f_{{6}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $+2\,f_{{1}}f_{{6}}a_{{0}}{a_{{1}}}^{2}a
_{{3}}$ $+4\,f_{{1}}f_{{6}}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $-2\,f_{{1}}f_{{6}}{a_{{1}}}^{3}a_{{2
}}$ $+30\,f_{{1}}f_{{7}}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $-18\,f_{{1}}f_{{7}}a_{{0}}a_{{1}}a_{{2}}
a_{{3}}$ $+f_{{1}}f_{{7}}a_{{0}}{a_{{2}}}^{3}$ $-2\,f_{{1}}f_{{7}}{a_{{1}}}^{3}a_{{3}}$ $-f_{{1}
}f_{{7}}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $-2\,{f_{{2}}}^{2}{a_{{0}}}^{3}a_{{2}}$ $-2\,f_{{2}}f_{{3
}}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $-2\,f_{{2}}f_{{4}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $+4\,f_{{2}}f_
{{4}}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $-2\,f_{{2}}f_{{4}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $+8\,f_{{2}}
f_{{5}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $-5\,f_{{2}}f_{{5}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $+f_{{2}}
f_{{5}}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $-f_{{2}}f_{{5}}{a_{{1}}}^{3}a_{{2}}$ $-2\,f_{{2}}f_{{6}}
a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $+2\,f_{{2}}f_{{6}}a_{{0}}{a_{{2}}}^{3}$ $-2\,f_{{2}}f_{{6}}{a_
{{1}}}^{2}{a_{{2}}}^{2}$ $-11\,f_{{2}}f_{{7}}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $+9\,f_{{2}}f_{{7}}
a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $-2\,f_{{2}}f_{{7}}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $-f_{{2}}f_{{7}}
a_{{1}}{a_{{2}}}^{3}$ $+6\,{f_{{3}}}^{2}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $-12\,f_{{3}}f_{{4}}{a_{{0
}}}^{2}a_{{2}}a_{{3}}$ $-2\,f_{{3}}f_{{4}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $+14\,f_{{3}}f_{{4}}a_
{{0}}a_{{1}}{a_{{2}}}^{2}$ $-72\,f_{{3}}f_{{5}}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $+28\,f_{{3}}f_{{5
}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $+4\,f_{{3}}f_{{5}}a_{{0}}{a_{{2}}}^{3}$ $+f_{{3}}f_{{5}}{a_{
{1}}}^{3}a_{{3}}$ $-f_{{3}}f_{{5}}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $+2\,f_{{3}}f_{{6}}{a_{{1}}}^{2
}a_{{2}}a_{{3}}$ $-2\,f_{{3}}f_{{6}}a_{{1}}{a_{{2}}}^{3}$ $+12\,f_{{3}}f_{{7}}a_{{0}}a_{{2}}{
a_{{3}}}^{2}$ $+2\,f_{{3}}f_{{7}}{a_{{1}}}^{2}{a_{{3}}}^{2}$ $-f_{{3}}f_{{7}}a_{{1}}{a_{{2}}}
^{2}a_{{3}}$ $-f_{{3}}f_{{7}}{a_{{2}}}^{4}$ $+6\,{f_{{4}}}^{2}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $-10\,
{f_{{4}}}^{2}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $-2\,{f_{{4}}}^{2}a_{{0}}{a_{{2}}}^{3}$ $-2\,{f_{{
4}}}^{2}{a_{{1}}}^{3}a_{{3}}$ $+8\,{f_{{4}}}^{2}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $-88\,f_{{4}}f_{{
5}}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $-6\,f_{{4}}f_{{5}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $+40\,f_{{4}}
f_{{5}}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $+20\,f_{{4}}f_{{6}}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $+2\,f_{
{4}}f_{{6}}{a_{{1}}}^{2}{a_{{3}}}^{2}$ $-2\,f_{{4}}f_{{6}}{a_{{2}}}^{4}$ $+78\,f_{{4}}f_{{7}}
a_{{0}}{a_{{3}}}^{3}$ $-4\,f_{{4}}f_{{7}}a_{{1}}a_{{2}}{a_{{3}}}^{2}$ $-8\,f_{{4}}f_{{7}}{a_{
{2}}}^{3}a_{{3}}$ $-80\,{f_{{5}}}^{2}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $+4\,{f_{{5}}}^{2}{a_{{1}}}
^{2}{a_{{3}}}^{2}$ $+30\,{f_{{5}}}^{2}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $-2\,{f_{{5}}}^{2}{a_{{2}}
}^{4}$ $-72\,f_{{5}}f_{{6}}a_{{0}}{a_{{3}}}^{3}$ $+60\,f_{{5}}f_{{6}}a_{{1}}a_{{2}}{a_{{3}}}^
{2}$ $-12\,f_{{5}}f_{{6}}{a_{{2}}}^{3}a_{{3}}$ $+46\,f_{{5}}f_{{7}}a_{{1}}{a_{{3}}}^{3}$ $-20\,f
_{{5}}f_{{7}}{a_{{2}}}^{2}{a_{{3}}}^{2}$ $-8\,{f_{{6}}}^{2}{a_{{2}}}^{2}{a_{{3}}}^{2}$ $-12\,
f_{{6}}f_{{7}}a_{{2}}{a_{{3}}}^{3}$ $+6\,{f_{{7}}}^{2}{a_{{3}}}^{4}$.

\noindent
${\bf eq[18]} = 
{a_{{0}}}^{2}{c_{{3}}}^{2}$ $-2\,a_{{0}}a_{{1}}c_{{2}}c_{{3}}$ $+4\,a_{{0}}a_{{3}}c_{{1}}c_{{
2}}$ $+{a_{{1}}}^{2}{c_{{2}}}^{2}
-4\,f_{{0}}{a_{{0}}}^{3}c_{{2}}$ $-4\,f_{{0}}{a_{{0}}}^{2}a_{{2}}c_{{0}}$ $+2\,f_{{1}}{a_{{0}
}}^{2}a_{{2}}c_{{1}}$ $-2\,f_{{1}}{a_{{0}}}^{2}a_{{3}}c_{{0}}$ $-2\,f_{{3}}{a_{{0}}}^{2}a_{{2
}}c_{{3}}$ $-4\,f_{{3}}{a_{{0}}}^{2}a_{{3}}c_{{2}}$ $+2\,f_{{3}}a_{{0}}a_{{1}}a_{{2}}c_{{2}}$ $+
2\,f_{{4}}{a_{{0}}}^{2}a_{{3}}c_{{3}}$ $-2\,f_{{4}}a_{{0}}a_{{1}}a_{{2}}c_{{3}}$ $-6\,f_{{4}}
a_{{0}}a_{{1}}a_{{3}}c_{{2}}$ $-4\,f_{{4}}a_{{0}}{a_{{3}}}^{2}c_{{0}}$ $+2\,f_{{4}}{a_{{1}}}^
{2}a_{{2}}c_{{2}}$ $-4\,f_{{5}}a_{{0}}a_{{2}}a_{{3}}c_{{2}}$ $+12\,f_{{5}}a_{{0}}{a_{{3}}}^{2
}c_{{1}}$ $-6\,f_{{5}}a_{{1}}a_{{2}}a_{{3}}c_{{1}}$ $-6\,f_{{5}}a_{{1}}{a_{{3}}}^{2}c_{{0}}$ $-4
\,f_{{6}}a_{{0}}{a_{{3}}}^{2}c_{{2}}$ $-2\,f_{{7}}a_{{0}}{a_{{3}}}^{2}c_{{3}}$ $-2\,f_{{7}}a_
{{1}}{a_{{3}}}^{2}c_{{2}}
-12\,f_{{0}}f_{{5}}{a_{{0}}}^{3}a_{{3}}$ $+4\,f_{{0}}f_{{5}}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $+4
\,f_{{0}}f_{{6}}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $+16\,f_{{0}}f_{{7}}{a_{{0}}}^{2}a_{{2}}a_{{3}
}$ $-4\,f_{{0}}f_{{7}}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $+{f_{{1}}}^{2}{a_{{0}}}^{4}$ $-2\,f_{{1}}f_{
{3}}{a_{{0}}}^{3}a_{{2}}$ $+2\,f_{{1}}f_{{4}}{a_{{0}}}^{3}a_{{3}}$ $-2\,f_{{1}}f_{{4}}{a_{{0}
}}^{2}a_{{1}}a_{{2}}$ $+2\,f_{{1}}f_{{5}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $-2\,f_{{1}}f_{{5}}{a_{
{0}}}^{2}{a_{{2}}}^{2}$ $+8\,f_{{1}}f_{{7}}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $-4\,f_{{1}}f_{{7}}a_{
{0}}a_{{1}}a_{{2}}a_{{3}}$ $+{f_{{3}}}^{2}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $-2\,f_{{3}}f_{{4}}{a_{
{0}}}^{2}a_{{2}}a_{{3}}$ $+2\,f_{{3}}f_{{4}}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $-12\,f_{{3}}f_{{5}}
{a_{{0}}}^{2}{a_{{3}}}^{2}$ $+6\,f_{{3}}f_{{5}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $+2\,f_{{3}}f_{{
7}}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $+{f_{{4}}}^{2}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $-2\,{f_{{4}}}^{2}
a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $+{f_{{4}}}^{2}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $-14\,f_{{4}}f_{{5}}
a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $+6\,f_{{4}}f_{{5}}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $+4\,f_{{4}}f_{{
6}}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $+14\,f_{{4}}f_{{7}}a_{{0}}{a_{{3}}}^{3}$ $-2\,f_{{4}}f_{{7}}
a_{{1}}a_{{2}}{a_{{3}}}^{2}$ $-12\,{f_{{5}}}^{2}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $-3\,{f_{{5}}}^{
2}{a_{{1}}}^{2}{a_{{3}}}^{2}$ $+6\,{f_{{5}}}^{2}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $-12\,f_{{5}}f_{
{6}}a_{{0}}{a_{{3}}}^{3}$ $+12\,f_{{5}}f_{{6}}a_{{1}}a_{{2}}{a_{{3}}}^{2}$ $+12\,f_{{5}}f_{{7
}}a_{{1}}{a_{{3}}}^{3}$ $+{f_{{7}}}^{2}{a_{{3}}}^{4}$.

\noindent
${\bf eq[19]} =
-7\,a_{{0}}a_{{1}}c_{{2}}c_{{3}}$ $+7\,a_{{0}}a_{{2}}c_{{1}}c_{{3}}$ $+a_{{0}}a_{{2}}{c_{{2}}
}^{2}$ $+18\,a_{{0}}a_{{3}}c_{{1}}c_{{2}}$ $-2\,{a_{{1}}}^{2}c_{{1}}c_{{3}}$ $+5\,{a_{{1}}}^{2}{
c_{{2}}}^{2}$ $-a_{{1}}a_{{2}}c_{{0}}c_{{3}}$ $+a_{{1}}a_{{3}}{c_{{1}}}^{2}$ $+2\,{a_{{2}}}^{2}c
_{{0}}c_{{2}}$ $-2\,{a_{{2}}}^{2}{c_{{1}}}^{2}
-16\,f_{{0}}{a_{{0}}}^{3}c_{{2}}$ $+2\,f_{{0}}{a_{{0}}}^{2}a_{{1}}c_{{1}}$ $-12\,f_{{0}}{a_{{0
}}}^{2}a_{{2}}c_{{0}}$ $-2\,f_{{0}}a_{{0}}{a_{{1}}}^{2}c_{{0}}$ $+f_{{1}}{a_{{0}}}^{2}a_{{1}}
c_{{2}}$ $+9\,f_{{1}}{a_{{0}}}^{2}a_{{2}}c_{{1}}$ $-4\,f_{{1}}{a_{{0}}}^{2}a_{{3}}c_{{0}}$ $-3\,
f_{{1}}a_{{0}}a_{{1}}a_{{2}}c_{{0}}$ $+f_{{1}}{a_{{1}}}^{3}c_{{0}}$ $+2\,f_{{2}}{a_{{0}}}^{2}
a_{{1}}c_{{3}}$ $-2\,f_{{2}}{a_{{0}}}^{2}a_{{3}}c_{{1}}$ $+f_{{2}}a_{{0}}a_{{1}}a_{{3}}c_{{0}
}$ $-2\,f_{{2}}a_{{0}}{a_{{2}}}^{2}c_{{0}}$ $+f_{{2}}{a_{{1}}}^{2}a_{{2}}c_{{0}}$ $-8\,f_{{3}}{a
_{{0}}}^{2}a_{{2}}c_{{3}}$ $-16\,f_{{3}}{a_{{0}}}^{2}a_{{3}}c_{{2}}$ $+2\,f_{{3}}a_{{0}}{a_{{
1}}}^{2}c_{{3}}$ $+7\,f_{{3}}a_{{0}}a_{{1}}a_{{2}}c_{{2}}$ $-f_{{3}}a_{{0}}{a_{{2}}}^{2}c_{{1
}}$ $-f_{{3}}{a_{{1}}}^{2}a_{{3}}c_{{0}}$ $+f_{{3}}a_{{1}}{a_{{2}}}^{2}c_{{0}}$ $+11\,f_{{4}}{a_
{{0}}}^{2}a_{{3}}c_{{3}}$ $-11\,f_{{4}}a_{{0}}a_{{1}}a_{{2}}c_{{3}}$ $-26\,f_{{4}}a_{{0}}a_{{
1}}a_{{3}}c_{{2}}$ $-15\,f_{{4}}a_{{0}}{a_{{3}}}^{2}c_{{0}}$ $+2\,f_{{4}}{a_{{1}}}^{3}c_{{3}}
+8\,f_{{4}}{a_{{1}}}^{2}a_{{2}}c_{{2}}$ $-3\,f_{{4}}a_{{1}}a_{{2}}a_{{3}}c_{{0}}$ $+2\,f_{{4}
}{a_{{2}}}^{3}c_{{0}}$ $-8\,f_{{5}}a_{{0}}{a_{{2}}}^{2}c_{{3}}$ $-14\,f_{{5}}a_{{0}}a_{{2}}a_
{{3}}c_{{2}}$ $+52\,f_{{5}}a_{{0}}{a_{{3}}}^{2}c_{{1}}$ $+3\,f_{{5}}{a_{{1}}}^{2}a_{{2}}c_{{3
}}$ $+7\,f_{{5}}{a_{{1}}}^{2}a_{{3}}c_{{2}}$ $-23\,f_{{5}}a_{{1}}a_{{2}}a_{{3}}c_{{1}}$ $-22\,f_
{{5}}a_{{1}}{a_{{3}}}^{2}c_{{0}}$ $+4\,f_{{5}}{a_{{2}}}^{3}c_{{1}}$ $+6\,f_{{5}}{a_{{2}}}^{2}
a_{{3}}c_{{0}}$ $-6\,f_{{6}}a_{{0}}a_{{2}}a_{{3}}c_{{3}}$ $-16\,f_{{6}}a_{{0}}{a_{{3}}}^{2}c_
{{2}}$ $+2\,f_{{6}}a_{{1}}{a_{{2}}}^{2}c_{{3}}$ $-2\,f_{{6}}a_{{1}}{a_{{3}}}^{2}c_{{1}}$ $-2\,f_
{{6}}{a_{{2}}}^{3}c_{{2}}$ $+4\,f_{{6}}{a_{{2}}}^{2}a_{{3}}c_{{1}}$ $+2\,f_{{6}}a_{{2}}{a_{{3
}}}^{2}c_{{0}}$ $-4\,f_{{7}}a_{{0}}{a_{{3}}}^{2}c_{{3}}$ $-9\,f_{{7}}a_{{1}}{a_{{3}}}^{2}c_{{
2}}$ $+f_{{7}}{a_{{2}}}^{3}c_{{3}}$ $-6\,f_{{7}}{a_{{2}}}^{2}a_{{3}}c_{{2}}$ $-f_{{7}}a_{{2}}{a_
{{3}}}^{2}c_{{1}}
-48\,f_{{0}}f_{{5}}{a_{{0}}}^{3}a_{{3}}$ $+12\,f_{{0}}f_{{5}}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $+2
\,f_{{0}}f_{{5}}a_{{0}}{a_{{1}}}^{3}$ $+8\,f_{{0}}f_{{6}}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $+4\,f_{
{0}}f_{{6}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $+52\,f_{{0}}f_{{7}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $+8
\,f_{{0}}f_{{7}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $-18\,f_{{0}}f_{{7}}a_{{0}}a_{{1}}{a_{{2}}}^{
2}$ $+4\,{f_{{1}}}^{2}{a_{{0}}}^{4}$ $-f_{{1}}f_{{2}}{a_{{0}}}^{3}a_{{1}}$ $-8\,f_{{1}}f_{{3}}{a
_{{0}}}^{3}a_{{2}}$ $-f_{{1}}f_{{3}}{a_{{0}}}^{2}{a_{{1}}}^{2}$ $+8\,f_{{1}}f_{{4}}{a_{{0}}}^
{3}a_{{3}}$ $-7\,f_{{1}}f_{{4}}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $-f_{{1}}f_{{4}}a_{{0}}{a_{{1}}}^
{3}$ $+10\,f_{{1}}f_{{5}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $-10\,f_{{1}}f_{{5}}{a_{{0}}}^{2}{a_{{2
}}}^{2}$ $+2\,f_{{1}}f_{{5}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $-f_{{1}}f_{{5}}{a_{{1}}}^{4}$ $-8\,f_{
{1}}f_{{6}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $+2\,f_{{1}}f_{{6}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $+4\,
f_{{1}}f_{{6}}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $-2\,f_{{1}}f_{{6}}{a_{{1}}}^{3}a_{{2}}$ $+20\,f_{
{1}}f_{{7}}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $-9\,f_{{1}}f_{{7}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $+f_{
{1}}f_{{7}}a_{{0}}{a_{{2}}}^{3}$ $-2\,f_{{1}}f_{{7}}{a_{{1}}}^{3}a_{{3}}$ $-f_{{1}}f_{{7}}{a_
{{1}}}^{2}{a_{{2}}}^{2}$ $-{f_{{2}}}^{2}{a_{{0}}}^{3}a_{{2}}$ $-f_{{2}}f_{{3}}{a_{{0}}}^{2}a_
{{1}}a_{{2}}$ $-f_{{2}}f_{{4}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $+2\,f_{{2}}f_{{4}}{a_{{0}}}^{2}{a
_{{2}}}^{2}$ $-f_{{2}}f_{{4}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $+2\,f_{{2}}f_{{5}}{a_{{0}}}^{2}a_{
{2}}a_{{3}}$ $-f_{{2}}f_{{5}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $+f_{{2}}f_{{5}}a_{{0}}a_{{1}}{a_{{
2}}}^{2}$ $-f_{{2}}f_{{5}}{a_{{1}}}^{3}a_{{2}}$ $+2\,f_{{2}}f_{{6}}a_{{0}}{a_{{2}}}^{3}$ $-2\,f_
{{2}}f_{{6}}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $-6\,f_{{2}}f_{{7}}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $+8\,
f_{{2}}f_{{7}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $-2\,f_{{2}}f_{{7}}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $-
f_{{2}}f_{{7}}a_{{1}}{a_{{2}}}^{3}$ $+4\,{f_{{3}}}^{2}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $-8\,f_{{3}
}f_{{4}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $-f_{{3}}f_{{4}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $+9\,f_{{3}
}f_{{4}}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $-48\,f_{{3}}f_{{5}}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $+20\,f_
{{3}}f_{{5}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $+2\,f_{{3}}f_{{5}}a_{{0}}{a_{{2}}}^{3}$ $+f_{{3}}f
_{{5}}{a_{{1}}}^{3}a_{{3}}$ $-f_{{3}}f_{{5}}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $+2\,f_{{3}}f_{{6}}{a
_{{1}}}^{2}a_{{2}}a_{{3}}$ $-2\,f_{{3}}f_{{6}}a_{{1}}{a_{{2}}}^{3}$ $+8\,f_{{3}}f_{{7}}a_{{0}
}a_{{2}}{a_{{3}}}^{2}$ $+2\,f_{{3}}f_{{7}}{a_{{1}}}^{2}{a_{{3}}}^{2}$ $-f_{{3}}f_{{7}}a_{{1}}
{a_{{2}}}^{2}a_{{3}}$ $-f_{{3}}f_{{7}}{a_{{2}}}^{4}$ $+4\,{f_{{4}}}^{2}{a_{{0}}}^{2}{a_{{3}}}
^{2}$ $-7\,{f_{{4}}}^{2}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $-{f_{{4}}}^{2}a_{{0}}{a_{{2}}}^{3}$ $-{f_
{{4}}}^{2}{a_{{1}}}^{3}a_{{3}}$ $+5\,{f_{{4}}}^{2}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $-60\,f_{{4}}f_
{{5}}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $-f_{{4}}f_{{5}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $+25\,f_{{4}}f
_{{5}}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $+14\,f_{{4}}f_{{6}}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $+2\,f_{{
4}}f_{{6}}{a_{{1}}}^{2}{a_{{3}}}^{2}$ $+2\,f_{{4}}f_{{6}}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $-2\,f_
{{4}}f_{{6}}{a_{{2}}}^{4}$ $+53\,f_{{4}}f_{{7}}a_{{0}}{a_{{3}}}^{3}$ $+2\,f_{{4}}f_{{7}}a_{{1
}}a_{{2}}{a_{{3}}}^{2}$ $-7\,f_{{4}}f_{{7}}{a_{{2}}}^{3}a_{{3}}$ $-48\,{f_{{5}}}^{2}a_{{0}}a_
{{2}}{a_{{3}}}^{2}$ $-2\,{f_{{5}}}^{2}{a_{{1}}}^{2}{a_{{3}}}^{2}$ $+22\,{f_{{5}}}^{2}a_{{1}}{
a_{{2}}}^{2}a_{{3}}$ $-2\,{f_{{5}}}^{2}{a_{{2}}}^{4}$ $-48\,f_{{5}}f_{{6}}a_{{0}}{a_{{3}}}^{3
}$ $+44\,f_{{5}}f_{{6}}a_{{1}}a_{{2}}{a_{{3}}}^{2}$ $-10\,f_{{5}}f_{{6}}{a_{{2}}}^{3}a_{{3}}$ $+
40\,f_{{5}}f_{{7}}a_{{1}}{a_{{3}}}^{3}$ $-18\,f_{{5}}f_{{7}}{a_{{2}}}^{2}{a_{{3}}}^{2}$ $-4\,
{f_{{6}}}^{2}{a_{{2}}}^{2}{a_{{3}}}^{2}$ $-6\,f_{{6}}f_{{7}}a_{{2}}{a_{{3}}}^{3}$ $+4\,{f_{{7
}}}^{2}{a_{{3}}}^{4}$.

\noindent
${\bf eq[20]} =
2\,a_{{0}}a_{{1}}{c_{{2}}}^{2}$ $+a_{{0}}a_{{2}}c_{{0}}c_{{3}}$ $+5\,a_{{0}}a_{{3}}c_{{0}}c_{
{2}}$ $+a_{{0}}a_{{3}}{c_{{1}}}^{2}$ $+2\,{a_{{1}}}^{2}c_{{1}}c_{{2}}$ $+a_{{1}}a_{{2}}{c_{{1}}}
^{2}$ $+{a_{{2}}}^{2}c_{{0}}c_{{1}}$ $+a_{{2}}a_{{3}}{c_{{0}}}^{2}
-2\,f_{{0}}{a_{{0}}}^{3}c_{{1}}$ $+2\,f_{{0}}{a_{{0}}}^{2}a_{{1}}c_{{0}}$ $-f_{{1}}{a_{{0}}}^
{3}c_{{2}}$ $+2\,f_{{1}}{a_{{0}}}^{2}a_{{2}}c_{{0}}$ $-f_{{1}}a_{{0}}{a_{{1}}}^{2}c_{{0}}$ $-2\,
f_{{2}}{a_{{0}}}^{2}a_{{1}}c_{{2}}$ $-f_{{2}}{a_{{0}}}^{2}a_{{3}}c_{{0}}$ $-f_{{2}}a_{{0}}a_{
{1}}a_{{2}}c_{{0}}$ $+f_{{3}}{a_{{0}}}^{2}a_{{2}}c_{{2}}$ $-2\,f_{{3}}{a_{{0}}}^{2}a_{{3}}c_{
{1}}$ $-2\,f_{{3}}a_{{0}}{a_{{1}}}^{2}c_{{2}}$ $+f_{{3}}a_{{0}}a_{{1}}a_{{3}}c_{{0}}$ $-2\,f_{{3
}}a_{{0}}{a_{{2}}}^{2}c_{{0}}$ $+3\,f_{{4}}{a_{{0}}}^{2}a_{{2}}c_{{3}}$ $-f_{{4}}{a_{{0}}}^{2
}a_{{3}}c_{{2}}$ $-2\,f_{{4}}a_{{0}}{a_{{1}}}^{2}c_{{3}}$ $-2\,f_{{4}}a_{{0}}a_{{1}}a_{{3}}c_
{{1}}$ $-3\,f_{{4}}a_{{0}}{a_{{2}}}^{2}c_{{1}}$ $+2\,f_{{4}}{a_{{1}}}^{2}a_{{2}}c_{{1}}$ $+2\,f_
{{4}}{a_{{1}}}^{2}a_{{3}}c_{{0}}$ $-f_{{4}}a_{{1}}{a_{{2}}}^{2}c_{{0}}$ $-2\,f_{{5}}{a_{{0}}}
^{2}a_{{3}}c_{{3}}$ $-4\,f_{{5}}a_{{0}}a_{{1}}a_{{2}}c_{{3}}$ $+11\,f_{{5}}a_{{0}}a_{{1}}a_{{
3}}c_{{2}}$ $-f_{{5}}a_{{0}}{a_{{2}}}^{2}c_{{2}}$ $+16\,f_{{5}}a_{{0}}{a_{{3}}}^{2}c_{{0}}$ $+f_
{{5}}{a_{{1}}}^{2}a_{{2}}c_{{2}}$ $+6\,f_{{5}}{a_{{1}}}^{2}a_{{3}}c_{{1}}$ $+f_{{5}}a_{{1}}a_
{{2}}a_{{3}}c_{{0}}$ $-f_{{5}}{a_{{2}}}^{3}c_{{0}}$ $-2\,f_{{6}}a_{{0}}{a_{{2}}}^{2}c_{{3}}$ $-8
\,f_{{6}}a_{{0}}a_{{2}}a_{{3}}c_{{2}}$ $-2\,f_{{6}}a_{{0}}{a_{{3}}}^{2}c_{{1}}$ $-2\,f_{{6}}{
a_{{2}}}^{3}c_{{1}}$ $-4\,f_{{6}}{a_{{2}}}^{2}a_{{3}}c_{{0}}$ $-4\,f_{{7}}a_{{0}}a_{{2}}a_{{3
}}c_{{3}}$ $-16\,f_{{7}}a_{{0}}{a_{{3}}}^{2}c_{{2}}$ $-2\,f_{{7}}a_{{1}}{a_{{3}}}^{2}c_{{1}}$ $+
f_{{7}}{a_{{2}}}^{3}c_{{2}}$ $-2\,f_{{7}}{a_{{2}}}^{2}a_{{3}}c_{{1}}$ $-6\,f_{{7}}a_{{2}}{a_{
{3}}}^{2}c_{{0}}
+
4\,f_{{0}}f_{{5}}{a_{{0}}}^{3}a_{{2}}$ $-2\,f_{{0}}f_{{5}}{a_{{0}}}^{2}{a_{{1}}}^{2}$ $-8\,f_
{{0}}f_{{7}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $-2\,f_{{0}}f_{{7}}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $+4\,
f_{{0}}f_{{7}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $+f_{{1}}f_{{2}}{a_{{0}}}^{4}$ $+f_{{1}}f_{{3}}{a_
{{0}}}^{3}a_{{1}}$ $-f_{{1}}f_{{4}}{a_{{0}}}^{3}a_{{2}}$ $+f_{{1}}f_{{4}}{a_{{0}}}^{2}{a_{{1}
}}^{2}$ $-2\,f_{{1}}f_{{5}}{a_{{0}}}^{3}a_{{3}}$ $-f_{{1}}f_{{5}}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $+
f_{{1}}f_{{5}}a_{{0}}{a_{{1}}}^{3}$ $+2\,f_{{1}}f_{{6}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $-4\,f_{{
1}}f_{{6}}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $+2\,f_{{1}}f_{{6}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $-7\,f_
{{1}}f_{{7}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $+6\,f_{{1}}f_{{7}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $-f_
{{1}}f_{{7}}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $-f_{{2}}f_{{3}}{a_{{0}}}^{3}a_{{2}}$ $+f_{{2}}f_{{4
}}{a_{{0}}}^{3}a_{{3}}$ $-f_{{2}}f_{{4}}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $-5\,f_{{2}}f_{{5}}{a_{{0
}}}^{2}a_{{1}}a_{{3}}$ $+f_{{2}}f_{{5}}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $+f_{{2}}f_{{5}}a_{{0}}{a_
{{1}}}^{2}a_{{2}}$ $+2\,f_{{2}}f_{{6}}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $+4\,f_{{2}}f_{{7}}{a_{{0}
}}^{2}{a_{{3}}}^{2}$ $+2\,f_{{2}}f_{{7}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $-f_{{2}}f_{{7}}a_{{0}}
{a_{{2}}}^{3}$ $-{f_{{3}}}^{2}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $+f_{{3}}f_{{4}}{a_{{0}}}^{2}a_{{1
}}a_{{3}}$ $+f_{{3}}f_{{4}}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $-2\,f_{{3}}f_{{4}}a_{{0}}{a_{{1}}}^{2
}a_{{2}}$ $+6\,f_{{3}}f_{{5}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $-7\,f_{{3}}f_{{5}}a_{{0}}{a_{{1}}}
^{2}a_{{3}}$ $+2\,f_{{3}}f_{{5}}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $-2\,f_{{3}}f_{{6}}a_{{0}}a_{{1}
}a_{{2}}a_{{3}}$ $+4\,f_{{3}}f_{{6}}a_{{0}}{a_{{2}}}^{3}$ $-4\,f_{{3}}f_{{7}}a_{{0}}a_{{1}}{a
_{{3}}}^{2}$ $+5\,f_{{3}}f_{{7}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $-{f_{{4}}}^{2}{a_{{0}}}^{2}a_{{
2}}a_{{3}}$ $+{f_{{4}}}^{2}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $+{f_{{4}}}^{2}a_{{0}}a_{{1}}{a_{{2}}
}^{2}$ $-{f_{{4}}}^{2}{a_{{1}}}^{3}a_{{2}}$ $-2\,f_{{4}}f_{{5}}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $+4\,
f_{{4}}f_{{5}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $+2\,f_{{4}}f_{{5}}a_{{0}}{a_{{2}}}^{3}$ $-2\,f_{
{4}}f_{{5}}{a_{{1}}}^{3}a_{{3}}$ $-f_{{4}}f_{{5}}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $+2\,f_{{4}}f_{{
6}}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $+2\,f_{{4}}f_{{6}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $-4\,f_{{4}}f
_{{6}}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $+2\,f_{{4}}f_{{6}}a_{{1}}{a_{{2}}}^{3}$ $-2\,f_{{4}}f_{{7
}}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $-4\,f_{{4}}f_{{7}}{a_{{1}}}^{2}{a_{{3}}}^{2}$ $+2\,f_{{4}}f_{
{7}}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $+f_{{4}}f_{{7}}{a_{{2}}}^{4}$ $+14\,{f_{{5}}}^{2}a_{{0}}a_{
{1}}{a_{{3}}}^{2}$ $-4\,{f_{{5}}}^{2}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $+{f_{{5}}}^{2}{a_{{1}}}^{2
}a_{{2}}a_{{3}}$ $-{f_{{5}}}^{2}a_{{1}}{a_{{2}}}^{3}$ $-24\,f_{{5}}f_{{6}}a_{{0}}a_{{2}}{a_{{
3}}}^{2}$ $-2\,f_{{5}}f_{{6}}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $+2\,f_{{5}}f_{{6}}{a_{{2}}}^{4}$ $-50
\,f_{{5}}f_{{7}}a_{{0}}{a_{{3}}}^{3}$ $-f_{{5}}f_{{7}}a_{{1}}a_{{2}}{a_{{3}}}^{2}$ $+5\,f_{{5
}}f_{{7}}{a_{{2}}}^{3}a_{{3}}$ $+4\,{f_{{6}}}^{2}{a_{{2}}}^{3}a_{{3}}$ $+12\,f_{{6}}f_{{7}}{a
_{{2}}}^{2}{a_{{3}}}^{2}$ $+8\,{f_{{7}}}^{2}a_{{2}}{a_{{3}}}^{3}$.

\noindent
${\bf eq[21]} =
{a_{{0}}}^{2}c_{{2}}c_{{3}}$ $-a_{{0}}a_{{1}}{c_{{2}}}^{2}$ $-2\,a_{{0}}a_{{3}}c_{{0}}c_{{2}}
-2\,a_{{0}}a_{{3}}{c_{{1}}}^{2}$ $+a_{{1}}a_{{2}}{c_{{1}}}^{2}$ $+a_{{1}}a_{{3}}c_{{0}}c_{{1}
}
+
2\,f_{{0}}{a_{{0}}}^{3}c_{{1}}$ $-2\,f_{{0}}{a_{{0}}}^{2}a_{{1}}c_{{0}}$ $+f_{{1}}{a_{{0}}}^{
3}c_{{2}}$ $+f_{{1}}{a_{{0}}}^{2}a_{{1}}c_{{1}}$ $-f_{{2}}{a_{{0}}}^{2}a_{{2}}c_{{1}}$ $+f_{{2}}
{a_{{0}}}^{2}a_{{3}}c_{{0}}$ $-2\,f_{{3}}{a_{{0}}}^{2}a_{{1}}c_{{3}}$ $-f_{{3}}{a_{{0}}}^{2}a
_{{2}}c_{{2}}$ $+2\,f_{{3}}{a_{{0}}}^{2}a_{{3}}c_{{1}}$ $+2\,f_{{3}}a_{{0}}{a_{{1}}}^{2}c_{{2
}}$ $+f_{{3}}a_{{0}}a_{{1}}a_{{3}}c_{{0}}$ $-f_{{4}}{a_{{0}}}^{2}a_{{2}}c_{{3}}$ $+f_{{4}}{a_{{0
}}}^{2}a_{{3}}c_{{2}}$ $+3\,f_{{4}}a_{{0}}a_{{1}}a_{{3}}c_{{1}}$ $+2\,f_{{4}}a_{{0}}{a_{{2}}}
^{2}c_{{1}}$ $-2\,f_{{4}}{a_{{1}}}^{2}a_{{2}}c_{{1}}$ $-f_{{4}}{a_{{1}}}^{2}a_{{3}}c_{{0}}$ $+8
\,f_{{5}}{a_{{0}}}^{2}a_{{3}}c_{{3}}$ $-2\,f_{{5}}a_{{0}}a_{{1}}a_{{2}}c_{{3}}$ $-9\,f_{{5}}a
_{{0}}a_{{1}}a_{{3}}c_{{2}}$ $-10\,f_{{5}}a_{{0}}{a_{{3}}}^{2}c_{{0}}$ $+2\,f_{{5}}{a_{{1}}}^
{2}a_{{2}}c_{{2}}$ $-f_{{5}}{a_{{1}}}^{2}a_{{3}}c_{{1}}$ $+f_{{5}}a_{{1}}a_{{2}}a_{{3}}c_{{0}
}$ $+2\,f_{{6}}a_{{0}}a_{{2}}a_{{3}}c_{{2}}$ $+2\,f_{{6}}a_{{0}}{a_{{3}}}^{2}c_{{1}}$ $-2\,f_{{6
}}a_{{1}}a_{{2}}a_{{3}}c_{{1}}$ $-2\,f_{{7}}a_{{0}}a_{{2}}a_{{3}}c_{{3}}$ $+7\,f_{{7}}a_{{0}}
{a_{{3}}}^{2}c_{{2}}$ $-2\,f_{{7}}a_{{1}}{a_{{3}}}^{2}c_{{1}}$ $+f_{{7}}{a_{{2}}}^{2}a_{{3}}c
_{{1}}$ $+f_{{7}}a_{{2}}{a_{{3}}}^{2}c_{{0}}
-4\,f_{{0}}f_{{5}}{a_{{0}}}^{3}a_{{2}}$ $+2\,f_{{0}}f_{{5}}{a_{{0}}}^{2}{a_{{1}}}^{2}$ $+8\,f
_{{0}}f_{{7}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $+2\,f_{{0}}f_{{7}}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $-4
\,f_{{0}}f_{{7}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $-f_{{1}}f_{{2}}{a_{{0}}}^{4}$ $-f_{{1}}f_{{3}}{
a_{{0}}}^{3}a_{{1}}$ $+f_{{1}}f_{{4}}{a_{{0}}}^{3}a_{{2}}$ $-f_{{1}}f_{{4}}{a_{{0}}}^{2}{a_{{
1}}}^{2}$ $+2\,f_{{1}}f_{{5}}{a_{{0}}}^{3}a_{{3}}$ $-f_{{1}}f_{{5}}{a_{{0}}}^{2}a_{{1}}a_{{2}
}$ $-2\,f_{{1}}f_{{6}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $+f_{{1}}f_{{7}}{a_{{0}}}^{2}a_{{2}}a_{{3}
}$ $-2\,f_{{1}}f_{{7}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $+f_{{2}}f_{{3}}{a_{{0}}}^{3}a_{{2}}$ $-f_{{2
}}f_{{4}}{a_{{0}}}^{3}a_{{3}}$ $+f_{{2}}f_{{4}}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $-f_{{2}}f_{{5}}{
a_{{0}}}^{2}a_{{1}}a_{{3}}$ $+f_{{2}}f_{{5}}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $-4\,f_{{2}}f_{{7}}{a
_{{0}}}^{2}{a_{{3}}}^{2}$ $+2\,f_{{2}}f_{{7}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $+{f_{{3}}}^{2}{a_
{{0}}}^{2}a_{{1}}a_{{2}}$ $-f_{{3}}f_{{4}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $-f_{{3}}f_{{4}}{a_{{0
}}}^{2}{a_{{2}}}^{2}$ $+2\,f_{{3}}f_{{4}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $-6\,f_{{3}}f_{{5}}{a_{
{0}}}^{2}a_{{2}}a_{{3}}$ $+5\,f_{{3}}f_{{5}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $-2\,f_{{3}}f_{{6}}a
_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $-2\,f_{{3}}f_{{7}}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $+f_{{3}}f_{{7}}
a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $+{f_{{4}}}^{2}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $-{f_{{4}}}^{2}a_{{0
}}{a_{{1}}}^{2}a_{{3}}$ $-{f_{{4}}}^{2}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $+{f_{{4}}}^{2}{a_{{1}}}^
{3}a_{{2}}$ $+2\,f_{{4}}f_{{5}}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $-4\,f_{{4}}f_{{5}}a_{{0}}a_{{1}}a
_{{2}}a_{{3}}$ $-2\,f_{{4}}f_{{5}}a_{{0}}{a_{{2}}}^{3}$ $+f_{{4}}f_{{5}}{a_{{1}}}^{3}a_{{3}}$ $+
2\,f_{{4}}f_{{5}}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $-2\,f_{{4}}f_{{6}}a_{{0}}a_{{1}}{a_{{3}}}^{2
}$ $-2\,f_{{4}}f_{{6}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $+2\,f_{{4}}f_{{6}}{a_{{1}}}^{2}a_{{2}}a_{
{3}}$ $+2\,f_{{4}}f_{{7}}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $+2\,f_{{4}}f_{{7}}{a_{{1}}}^{2}{a_{{3}
}}^{2}$ $-f_{{4}}f_{{7}}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $-14\,{f_{{5}}}^{2}a_{{0}}a_{{1}}{a_{{3}
}}^{2}$ $+2\,{f_{{5}}}^{2}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $+5\,{f_{{5}}}^{2}{a_{{1}}}^{2}a_{{2}}
a_{{3}}$ $-{f_{{5}}}^{2}a_{{1}}{a_{{2}}}^{3}$ $+12\,f_{{5}}f_{{6}}a_{{0}}a_{{2}}{a_{{3}}}^{2}
-2\,f_{{5}}f_{{6}}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $+32\,f_{{5}}f_{{7}}a_{{0}}{a_{{3}}}^{3}$ $-5
\,f_{{5}}f_{{7}}a_{{1}}a_{{2}}{a_{{3}}}^{2}$ $-f_{{5}}f_{{7}}{a_{{2}}}^{3}a_{{3}}$ $-2\,f_{{6
}}f_{{7}}{a_{{2}}}^{2}{a_{{3}}}^{2}$ $-2\,{f_{{7}}}^{2}a_{{2}}{a_{{3}}}^{3}$.

\noindent
${\bf eq[22]} =
-2\,a_{{0}}a_{{1}}{c_{{2}}}^{2}$ $-2\,a_{{0}}a_{{2}}c_{{0}}c_{{3}}$ $-5\,a_{{0}}a_{{3}}c_{{0}
}c_{{2}}$ $-a_{{0}}a_{{3}}{c_{{1}}}^{2}$ $-2\,{a_{{1}}}^{2}c_{{1}}c_{{2}}$ $+a_{{1}}a_{{2}}c_{{0
}}c_{{2}}$ $-a_{{1}}a_{{2}}{c_{{1}}}^{2}
+
2\,f_{{0}}{a_{{0}}}^{3}c_{{1}}$ $-2\,f_{{0}}{a_{{0}}}^{2}a_{{1}}c_{{0}}$ $+f_{{1}}{a_{{0}}}^{
3}c_{{2}}$ $-2\,f_{{1}}{a_{{0}}}^{2}a_{{2}}c_{{0}}$ $+f_{{1}}a_{{0}}{a_{{1}}}^{2}c_{{0}}$ $+2\,f
_{{2}}{a_{{0}}}^{2}a_{{1}}c_{{2}}$ $+f_{{2}}{a_{{0}}}^{2}a_{{3}}c_{{0}}$ $+f_{{2}}a_{{0}}a_{{
1}}a_{{2}}c_{{0}}$ $-f_{{3}}{a_{{0}}}^{2}a_{{2}}c_{{2}}$ $+2\,f_{{3}}{a_{{0}}}^{2}a_{{3}}c_{{
1}}$ $+2\,f_{{3}}a_{{0}}{a_{{1}}}^{2}c_{{2}}$ $-f_{{3}}a_{{0}}a_{{1}}a_{{3}}c_{{0}}$ $+2\,f_{{3}
}a_{{0}}{a_{{2}}}^{2}c_{{0}}$ $-3\,f_{{4}}{a_{{0}}}^{2}a_{{2}}c_{{3}}$ $+f_{{4}}{a_{{0}}}^{2}
a_{{3}}c_{{2}}$ $+2\,f_{{4}}a_{{0}}{a_{{1}}}^{2}c_{{3}}$ $+2\,f_{{4}}a_{{0}}a_{{1}}a_{{3}}c_{
{1}}$ $+3\,f_{{4}}a_{{0}}{a_{{2}}}^{2}c_{{1}}$ $-2\,f_{{4}}{a_{{1}}}^{2}a_{{2}}c_{{1}}$ $-2\,f_{
{4}}{a_{{1}}}^{2}a_{{3}}c_{{0}}$ $+f_{{4}}a_{{1}}{a_{{2}}}^{2}c_{{0}}$ $+2\,f_{{5}}{a_{{0}}}^
{2}a_{{3}}c_{{3}}$ $+4\,f_{{5}}a_{{0}}a_{{1}}a_{{2}}c_{{3}}$ $-11\,f_{{5}}a_{{0}}a_{{1}}a_{{3
}}c_{{2}}$ $+f_{{5}}a_{{0}}{a_{{2}}}^{2}c_{{2}}$ $-16\,f_{{5}}a_{{0}}{a_{{3}}}^{2}c_{{0}}$ $-f_{
{5}}{a_{{1}}}^{2}a_{{2}}c_{{2}}$ $-6\,f_{{5}}{a_{{1}}}^{2}a_{{3}}c_{{1}}$ $+f_{{5}}a_{{1}}a_{
{2}}a_{{3}}c_{{0}}$ $+2\,f_{{6}}a_{{0}}{a_{{2}}}^{2}c_{{3}}$ $+8\,f_{{6}}a_{{0}}a_{{2}}a_{{3}
}c_{{2}}$ $+2\,f_{{6}}a_{{0}}{a_{{3}}}^{2}c_{{1}}$ $+2\,f_{{6}}{a_{{2}}}^{3}c_{{1}}$ $+2\,f_{{6}
}{a_{{2}}}^{2}a_{{3}}c_{{0}}$ $+4\,f_{{7}}a_{{0}}a_{{2}}a_{{3}}c_{{3}}$ $+16\,f_{{7}}a_{{0}}{
a_{{3}}}^{2}c_{{2}}$ $+2\,f_{{7}}a_{{1}}{a_{{3}}}^{2}c_{{1}}$ $-f_{{7}}{a_{{2}}}^{3}c_{{2}}$ $+2
\,f_{{7}}{a_{{2}}}^{2}a_{{3}}c_{{1}}$ $+3\,f_{{7}}a_{{2}}{a_{{3}}}^{2}c_{{0}}
-4\,f_{{0}}f_{{5}}{a_{{0}}}^{3}a_{{2}}$ $+2\,f_{{0}}f_{{5}}{a_{{0}}}^{2}{a_{{1}}}^{2}$ $+8\,f
_{{0}}f_{{7}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $+2\,f_{{0}}f_{{7}}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $-4
\,f_{{0}}f_{{7}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $-f_{{1}}f_{{2}}{a_{{0}}}^{4}$ $-f_{{1}}f_{{3}}{
a_{{0}}}^{3}a_{{1}}$ $+f_{{1}}f_{{4}}{a_{{0}}}^{3}a_{{2}}$ $-f_{{1}}f_{{4}}{a_{{0}}}^{2}{a_{{
1}}}^{2}$ $+2\,f_{{1}}f_{{5}}{a_{{0}}}^{3}a_{{3}}$ $+f_{{1}}f_{{5}}{a_{{0}}}^{2}a_{{1}}a_{{2}
}$ $-f_{{1}}f_{{5}}a_{{0}}{a_{{1}}}^{3}$ $-2\,f_{{1}}f_{{6}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $+4\,f_
{{1}}f_{{6}}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $-2\,f_{{1}}f_{{6}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $+7\,
f_{{1}}f_{{7}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $-6\,f_{{1}}f_{{7}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $+
f_{{1}}f_{{7}}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $+f_{{2}}f_{{3}}{a_{{0}}}^{3}a_{{2}}$ $-f_{{2}}f_{
{4}}{a_{{0}}}^{3}a_{{3}}$ $+f_{{2}}f_{{4}}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $+5\,f_{{2}}f_{{5}}{a_
{{0}}}^{2}a_{{1}}a_{{3}}$ $-f_{{2}}f_{{5}}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $-f_{{2}}f_{{5}}a_{{0}}
{a_{{1}}}^{2}a_{{2}}$ $-2\,f_{{2}}f_{{6}}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $-4\,f_{{2}}f_{{7}}{a_{
{0}}}^{2}{a_{{3}}}^{2}$ $-2\,f_{{2}}f_{{7}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $+f_{{2}}f_{{7}}a_{{0
}}{a_{{2}}}^{3}$ $+{f_{{3}}}^{2}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $-f_{{3}}f_{{4}}{a_{{0}}}^{2}a_{
{1}}a_{{3}}$ $-f_{{3}}f_{{4}}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $+2\,f_{{3}}f_{{4}}a_{{0}}{a_{{1}}}^
{2}a_{{2}}$ $-6\,f_{{3}}f_{{5}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $+7\,f_{{3}}f_{{5}}a_{{0}}{a_{{1}
}}^{2}a_{{3}}$ $-2\,f_{{3}}f_{{5}}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $+2\,f_{{3}}f_{{6}}a_{{0}}a_{{
1}}a_{{2}}a_{{3}}$ $-4\,f_{{3}}f_{{6}}a_{{0}}{a_{{2}}}^{3}$ $+4\,f_{{3}}f_{{7}}a_{{0}}a_{{1}}
{a_{{3}}}^{2}$ $-5\,f_{{3}}f_{{7}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $+{f_{{4}}}^{2}{a_{{0}}}^{2}a_
{{2}}a_{{3}}$ $-{f_{{4}}}^{2}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $-{f_{{4}}}^{2}a_{{0}}a_{{1}}{a_{{2
}}}^{2}$ $+{f_{{4}}}^{2}{a_{{1}}}^{3}a_{{2}}$ $+2\,f_{{4}}f_{{5}}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $-4
\,f_{{4}}f_{{5}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $-2\,f_{{4}}f_{{5}}a_{{0}}{a_{{2}}}^{3}$ $+2\,f
_{{4}}f_{{5}}{a_{{1}}}^{3}a_{{3}}$ $+f_{{4}}f_{{5}}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $-2\,f_{{4}}f_
{{6}}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $-2\,f_{{4}}f_{{6}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $+4\,f_{{4}
}f_{{6}}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $-2\,f_{{4}}f_{{6}}a_{{1}}{a_{{2}}}^{3}$ $+2\,f_{{4}}f_{
{7}}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $+4\,f_{{4}}f_{{7}}{a_{{1}}}^{2}{a_{{3}}}^{2}$ $-2\,f_{{4}}f
_{{7}}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $-f_{{4}}f_{{7}}{a_{{2}}}^{4}$ $-14\,{f_{{5}}}^{2}a_{{0}}a
_{{1}}{a_{{3}}}^{2}$ $+4\,{f_{{5}}}^{2}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $-{f_{{5}}}^{2}{a_{{1}}}^
{2}a_{{2}}a_{{3}}$ $+{f_{{5}}}^{2}a_{{1}}{a_{{2}}}^{3}$ $+24\,f_{{5}}f_{{6}}a_{{0}}a_{{2}}{a_
{{3}}}^{2}$ $+2\,f_{{5}}f_{{6}}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $-2\,f_{{5}}f_{{6}}{a_{{2}}}^{4}$ $+
50\,f_{{5}}f_{{7}}a_{{0}}{a_{{3}}}^{3}$ $+f_{{5}}f_{{7}}a_{{1}}a_{{2}}{a_{{3}}}^{2}$ $-5\,f_{
{5}}f_{{7}}{a_{{2}}}^{3}a_{{3}}$ $-4\,{f_{{6}}}^{2}{a_{{2}}}^{3}a_{{3}}$ $-12\,f_{{6}}f_{{7}}
{a_{{2}}}^{2}{a_{{3}}}^{2}$ $-8\,{f_{{7}}}^{2}a_{{2}}{a_{{3}}}^{3}$.

\noindent
${\bf eq[23]} = 
a_{{0}}a_{{2}}c_{{1}}c_{{2}}$ $-a_{{0}}a_{{3}}c_{{0}}c_{{2}}$ $-2\,a_{{0}}a_{{3}}{c_{{1}}}^{2
}$ $+a_{{1}}a_{{2}}{c_{{1}}}^{2}$ $+a_{{1}}a_{{3}}c_{{0}}c_{{1}}
+
2\,f_{{0}}{a_{{0}}}^{3}c_{{1}}$ $-2\,f_{{0}}{a_{{0}}}^{2}a_{{1}}c_{{0}}$ $+f_{{1}}{a_{{0}}}^{
3}c_{{2}}$ $+f_{{1}}{a_{{0}}}^{2}a_{{1}}c_{{1}}$ $-f_{{2}}{a_{{0}}}^{2}a_{{2}}c_{{1}}$ $+f_{{2}}
{a_{{0}}}^{2}a_{{3}}c_{{0}}$ $-2\,f_{{3}}{a_{{0}}}^{2}a_{{1}}c_{{3}}$ $-f_{{3}}{a_{{0}}}^{2}a
_{{2}}c_{{2}}$ $+2\,f_{{3}}{a_{{0}}}^{2}a_{{3}}c_{{1}}$ $+2\,f_{{3}}a_{{0}}{a_{{1}}}^{2}c_{{2
}}$ $+f_{{3}}a_{{0}}a_{{1}}a_{{3}}c_{{0}}$ $-f_{{4}}{a_{{0}}}^{2}a_{{2}}c_{{3}}$ $+f_{{4}}{a_{{0
}}}^{2}a_{{3}}c_{{2}}$ $+3\,f_{{4}}a_{{0}}a_{{1}}a_{{3}}c_{{1}}$ $+2\,f_{{4}}a_{{0}}{a_{{2}}}
^{2}c_{{1}}$ $-2\,f_{{4}}{a_{{1}}}^{2}a_{{2}}c_{{1}}$ $-f_{{4}}{a_{{1}}}^{2}a_{{3}}c_{{0}}$ $+8
\,f_{{5}}{a_{{0}}}^{2}a_{{3}}c_{{3}}$ $-2\,f_{{5}}a_{{0}}a_{{1}}a_{{2}}c_{{3}}$ $-7\,f_{{5}}a
_{{0}}a_{{1}}a_{{3}}c_{{2}}$ $-f_{{5}}a_{{0}}{a_{{2}}}^{2}c_{{2}}$ $-10\,f_{{5}}a_{{0}}{a_{{3
}}}^{2}c_{{0}}$ $+2\,f_{{5}}{a_{{1}}}^{2}a_{{2}}c_{{2}}$ $-f_{{5}}{a_{{1}}}^{2}a_{{3}}c_{{1}}
+f_{{5}}a_{{1}}a_{{2}}a_{{3}}c_{{0}}$ $+2\,f_{{6}}a_{{0}}{a_{{3}}}^{2}c_{{1}}$ $-2\,f_{{6}}a_
{{1}}a_{{2}}a_{{3}}c_{{1}}$ $-2\,f_{{7}}a_{{0}}a_{{2}}a_{{3}}c_{{3}}$ $+4\,f_{{7}}a_{{0}}{a_{
{3}}}^{2}c_{{2}}$ $-2\,f_{{7}}a_{{1}}{a_{{3}}}^{2}c_{{1}}$ $+f_{{7}}{a_{{2}}}^{2}a_{{3}}c_{{1
}}$ $+f_{{7}}a_{{2}}{a_{{3}}}^{2}c_{{0}}
-4\,f_{{0}}f_{{5}}{a_{{0}}}^{3}a_{{2}}$ $+2\,f_{{0}}f_{{5}}{a_{{0}}}^{2}{a_{{1}}}^{2}$ $+8\,f
_{{0}}f_{{7}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $+2\,f_{{0}}f_{{7}}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $-4
\,f_{{0}}f_{{7}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $-f_{{1}}f_{{2}}{a_{{0}}}^{4}$ $-f_{{1}}f_{{3}}{
a_{{0}}}^{3}a_{{1}}$ $+f_{{1}}f_{{4}}{a_{{0}}}^{3}a_{{2}}$ $-f_{{1}}f_{{4}}{a_{{0}}}^{2}{a_{{
1}}}^{2}$ $+2\,f_{{1}}f_{{5}}{a_{{0}}}^{3}a_{{3}}$ $-f_{{1}}f_{{5}}{a_{{0}}}^{2}a_{{1}}a_{{2}
}$ $-2\,f_{{1}}f_{{6}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $+f_{{1}}f_{{7}}{a_{{0}}}^{2}a_{{2}}a_{{3}
}$ $-2\,f_{{1}}f_{{7}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $+f_{{2}}f_{{3}}{a_{{0}}}^{3}a_{{2}}$ $-f_{{2
}}f_{{4}}{a_{{0}}}^{3}a_{{3}}$ $+f_{{2}}f_{{4}}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $-f_{{2}}f_{{5}}{
a_{{0}}}^{2}a_{{1}}a_{{3}}$ $+f_{{2}}f_{{5}}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $-4\,f_{{2}}f_{{7}}{a
_{{0}}}^{2}{a_{{3}}}^{2}$ $+2\,f_{{2}}f_{{7}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $+{f_{{3}}}^{2}{a_
{{0}}}^{2}a_{{1}}a_{{2}}$ $-f_{{3}}f_{{4}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $-f_{{3}}f_{{4}}{a_{{0
}}}^{2}{a_{{2}}}^{2}$ $+2\,f_{{3}}f_{{4}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $-6\,f_{{3}}f_{{5}}{a_{
{0}}}^{2}a_{{2}}a_{{3}}$ $+5\,f_{{3}}f_{{5}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $-2\,f_{{3}}f_{{6}}a
_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $-2\,f_{{3}}f_{{7}}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $+f_{{3}}f_{{7}}
a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $+{f_{{4}}}^{2}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $-{f_{{4}}}^{2}a_{{0
}}{a_{{1}}}^{2}a_{{3}}$ $-{f_{{4}}}^{2}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $+{f_{{4}}}^{2}{a_{{1}}}^
{3}a_{{2}}$ $+2\,f_{{4}}f_{{5}}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $-4\,f_{{4}}f_{{5}}a_{{0}}a_{{1}}a
_{{2}}a_{{3}}$ $-2\,f_{{4}}f_{{5}}a_{{0}}{a_{{2}}}^{3}$ $+f_{{4}}f_{{5}}{a_{{1}}}^{3}a_{{3}}$ $+
2\,f_{{4}}f_{{5}}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $-2\,f_{{4}}f_{{6}}a_{{0}}a_{{1}}{a_{{3}}}^{2
}$ $-2\,f_{{4}}f_{{6}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $+2\,f_{{4}}f_{{6}}{a_{{1}}}^{2}a_{{2}}a_{
{3}}$ $+2\,f_{{4}}f_{{7}}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $+2\,f_{{4}}f_{{7}}{a_{{1}}}^{2}{a_{{3}
}}^{2}$ $-f_{{4}}f_{{7}}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $-14\,{f_{{5}}}^{2}a_{{0}}a_{{1}}{a_{{3}
}}^{2}$ $+2\,{f_{{5}}}^{2}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $+5\,{f_{{5}}}^{2}{a_{{1}}}^{2}a_{{2}}
a_{{3}}$ $-{f_{{5}}}^{2}a_{{1}}{a_{{2}}}^{3}$ $+12\,f_{{5}}f_{{6}}a_{{0}}a_{{2}}{a_{{3}}}^{2}
-2\,f_{{5}}f_{{6}}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $+32\,f_{{5}}f_{{7}}a_{{0}}{a_{{3}}}^{3}$ $-5
\,f_{{5}}f_{{7}}a_{{1}}a_{{2}}{a_{{3}}}^{2}$ $-f_{{5}}f_{{7}}{a_{{2}}}^{3}a_{{3}}$ $-2\,f_{{6
}}f_{{7}}{a_{{2}}}^{2}{a_{{3}}}^{2}$ $-2\,{f_{{7}}}^{2}a_{{2}}{a_{{3}}}^{3}$.

\noindent
${\bf eq[24]} =
{a_{{0}}}^{2}c_{{1}}c_{{3}}$ $+a_{{0}}a_{{1}}c_{{0}}c_{{3}}$ $-3\,a_{{0}}a_{{1}}c_{{1}}c_{{2}
}$ $+2\,a_{{0}}a_{{2}}c_{{0}}c_{{2}}$ $-10\,a_{{0}}a_{{3}}c_{{0}}c_{{1}}$ $-{a_{{1}}}^{2}c_{{0}}
c_{{2}}$ $-2\,{a_{{1}}}^{2}{c_{{1}}}^{2}
+
8\,f_{{0}}{a_{{0}}}^{3}c_{{0}}$ $+f_{{1}}{a_{{0}}}^{3}c_{{1}}$ $-f_{{1}}{a_{{0}}}^{2}a_{{1}}c
_{{0}}$ $+2\,f_{{2}}{a_{{0}}}^{3}c_{{2}}$ $+4\,f_{{2}}{a_{{0}}}^{2}a_{{1}}c_{{1}}$ $-2\,f_{{2}}{
a_{{0}}}^{2}a_{{2}}c_{{0}}$ $+2\,f_{{3}}{a_{{0}}}^{2}a_{{1}}c_{{2}}$ $-f_{{3}}{a_{{0}}}^{2}a_
{{2}}c_{{1}}$ $+8\,f_{{3}}{a_{{0}}}^{2}a_{{3}}c_{{0}}$ $+4\,f_{{3}}a_{{0}}{a_{{1}}}^{2}c_{{1}
}$ $-f_{{3}}a_{{0}}a_{{1}}a_{{2}}c_{{0}}$ $-5\,f_{{4}}{a_{{0}}}^{2}a_{{1}}c_{{3}}$ $-2\,f_{{4}}{
a_{{0}}}^{2}a_{{2}}c_{{2}}$ $+f_{{4}}{a_{{0}}}^{2}a_{{3}}c_{{1}}$ $+7\,f_{{4}}a_{{0}}{a_{{1}}
}^{2}c_{{2}}$ $+12\,f_{{4}}a_{{0}}a_{{1}}a_{{3}}c_{{0}}$ $+2\,f_{{4}}a_{{0}}{a_{{2}}}^{2}c_{{0
}}$ $+4\,f_{{4}}{a_{{1}}}^{3}c_{{1}}$ $-f_{{4}}{a_{{1}}}^{2}a_{{2}}c_{{0}}$ $+14\,f_{{5}}{a_{{0}
}}^{2}a_{{2}}c_{{3}}$ $+2\,f_{{5}}{a_{{0}}}^{2}a_{{3}}c_{{2}}$ $-14\,f_{{5}}a_{{0}}a_{{1}}a_{
{2}}c_{{2}}$ $-3\,f_{{5}}a_{{0}}a_{{1}}a_{{3}}c_{{1}}$ $-16\,f_{{5}}a_{{0}}{a_{{2}}}^{2}c_{{1
}}$ $+3\,f_{{5}}{a_{{1}}}^{2}a_{{2}}c_{{1}}$ $-4\,f_{{5}}{a_{{1}}}^{2}a_{{3}}c_{{0}}$ $+18\,f_{{
6}}{a_{{0}}}^{2}a_{{3}}c_{{3}}$ $-2\,f_{{6}}a_{{0}}a_{{1}}a_{{2}}c_{{3}}$ $-14\,f_{{6}}a_{{0}
}a_{{1}}a_{{3}}c_{{2}}$ $-2\,f_{{6}}a_{{0}}{a_{{2}}}^{2}c_{{2}}$ $-10\,f_{{6}}a_{{0}}{a_{{3}}
}^{2}c_{{0}}$ $+2\,f_{{6}}{a_{{1}}}^{2}a_{{2}}c_{{2}}$ $+2\,f_{{7}}a_{{0}}{a_{{2}}}^{2}c_{{3}
}$ $-10\,f_{{7}}a_{{0}}a_{{2}}a_{{3}}c_{{2}}$ $+31\,f_{{7}}a_{{0}}{a_{{3}}}^{2}c_{{1}}$ $+4\,f_{
{7}}{a_{{1}}}^{2}a_{{3}}c_{{2}}$ $-2\,f_{{7}}a_{{1}}a_{{2}}a_{{3}}c_{{1}}$ $+3\,f_{{7}}a_{{1}
}{a_{{3}}}^{2}c_{{0}}$ $-f_{{7}}{a_{{2}}}^{3}c_{{1}}$ $-f_{{7}}{a_{{2}}}^{2}a_{{3}}c_{{0}}
-16\,f_{{0}}f_{{6}}{a_{{0}}}^{3}a_{{2}}$ $+8\,f_{{0}}f_{{6}}{a_{{0}}}^{2}{a_{{1}}}^{2}$ $-24
\,f_{{0}}f_{{7}}{a_{{0}}}^{3}a_{{3}}$ $-8\,f_{{0}}f_{{7}}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $+8\,f_
{{0}}f_{{7}}a_{{0}}{a_{{1}}}^{3}$ $-2\,f_{{1}}f_{{5}}{a_{{0}}}^{3}a_{{2}}$ $+f_{{1}}f_{{5}}{a
_{{0}}}^{2}{a_{{1}}}^{2}$ $+4\,f_{{1}}f_{{7}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $+f_{{1}}f_{{7}}{a_
{{0}}}^{2}{a_{{2}}}^{2}$ $-2\,f_{{1}}f_{{7}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $-2\,{f_{{2}}}^{2}{a
_{{0}}}^{4}$ $-4\,f_{{2}}f_{{3}}{a_{{0}}}^{3}a_{{1}}$ $+4\,f_{{2}}f_{{4}}{a_{{0}}}^{3}a_{{2}}
-4\,f_{{2}}f_{{4}}{a_{{0}}}^{2}{a_{{1}}}^{2}$ $+2\,f_{{2}}f_{{5}}{a_{{0}}}^{3}a_{{3}}$ $-2\,f
_{{2}}f_{{5}}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $-8\,f_{{2}}f_{{6}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $+2
\,f_{{2}}f_{{6}}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $+12\,f_{{2}}f_{{7}}{a_{{0}}}^{2}a_{{2}}a_{{3}
}$ $-8\,f_{{2}}f_{{7}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $-2\,f_{{2}}f_{{7}}a_{{0}}a_{{1}}{a_{{2}}}
^{2}$ $-2\,{f_{{3}}}^{2}{a_{{0}}}^{2}{a_{{1}}}^{2}$ $+4\,f_{{3}}f_{{4}}{a_{{0}}}^{2}a_{{1}}a_
{{2}}$ $-4\,f_{{3}}f_{{4}}a_{{0}}{a_{{1}}}^{3}$ $+2\,f_{{3}}f_{{5}}{a_{{0}}}^{2}a_{{1}}a_{{3}
}$ $+2\,f_{{3}}f_{{5}}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $-3\,f_{{3}}f_{{5}}a_{{0}}{a_{{1}}}^{2}a_{{
2}}$ $-16\,f_{{3}}f_{{6}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $+2\,f_{{3}}f_{{6}}a_{{0}}a_{{1}}{a_{{2
}}}^{2}$ $-24\,f_{{3}}f_{{7}}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $-f_{{3}}f_{{7}}a_{{0}}{a_{{2}}}^{3}
-2\,{f_{{4}}}^{2}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $+4\,{f_{{4}}}^{2}a_{{0}}{a_{{1}}}^{2}a_{{2}}
-2\,{f_{{4}}}^{2}{a_{{1}}}^{4}$ $-4\,f_{{4}}f_{{5}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $+13\,f_{{4}}
f_{{5}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $-f_{{4}}f_{{5}}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $-3\,f_{{4}}
f_{{5}}{a_{{1}}}^{3}a_{{2}}$ $-18\,f_{{4}}f_{{6}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $-2\,f_{{4}}f_
{{6}}a_{{0}}{a_{{2}}}^{3}$ $+2\,f_{{4}}f_{{6}}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $-35\,f_{{4}}f_{{7}
}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $-11\,f_{{4}}f_{{7}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $+6\,f_{{4}}f_
{{7}}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $+f_{{4}}f_{{7}}a_{{1}}{a_{{2}}}^{3}$ $+4\,{f_{{5}}}^{2}{a_
{{0}}}^{2}{a_{{3}}}^{2}$ $-32\,{f_{{5}}}^{2}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $+16\,{f_{{5}}}^{2}
a_{{0}}{a_{{2}}}^{3}$ $+{f_{{5}}}^{2}{a_{{1}}}^{3}a_{{3}}$ $-{f_{{5}}}^{2}{a_{{1}}}^{2}{a_{{2
}}}^{2}$ $-32\,f_{{5}}f_{{6}}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $+26\,f_{{5}}f_{{6}}a_{{0}}{a_{{2}}
}^{2}a_{{3}}$ $+8\,f_{{5}}f_{{6}}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $-8\,f_{{5}}f_{{7}}a_{{0}}a_{{2
}}{a_{{3}}}^{2}$ $+12\,f_{{5}}f_{{7}}{a_{{1}}}^{2}{a_{{3}}}^{2}$ $+3\,f_{{5}}f_{{7}}a_{{1}}{a
_{{2}}}^{2}a_{{3}}$ $+f_{{5}}f_{{7}}{a_{{2}}}^{4}$ $+20\,{f_{{6}}}^{2}a_{{0}}a_{{2}}{a_{{3}}}
^{2}$ $+30\,f_{{6}}f_{{7}}a_{{0}}{a_{{3}}}^{3}$ $-6\,f_{{6}}f_{{7}}a_{{1}}a_{{2}}{a_{{3}}}^{2
}$ $+2\,f_{{6}}f_{{7}}{a_{{2}}}^{3}a_{{3}}$ $-8\,{f_{{7}}}^{2}a_{{1}}{a_{{3}}}^{3}$.

\noindent
${\bf eq[25]} =
{a_{{0}}}^{2}{c_{{2}}}^{2}$ $+2\,a_{{0}}a_{{1}}c_{{1}}c_{{2}}$ $+4\,a_{{0}}a_{{3}}c_{{0}}c_{{
1}}$ $+{a_{{1}}}^{2}{c_{{1}}}^{2}
-4\,f_{{0}}{a_{{0}}}^{3}c_{{0}}$ $-2\,f_{{2}}{a_{{0}}}^{3}c_{{2}}$ $-2\,f_{{2}}{a_{{0}}}^{2}a
_{{1}}c_{{1}}$ $-2\,f_{{3}}{a_{{0}}}^{2}a_{{1}}c_{{2}}$ $-4\,f_{{3}}{a_{{0}}}^{2}a_{{3}}c_{{0
}}$ $-2\,f_{{3}}a_{{0}}{a_{{1}}}^{2}c_{{1}}$ $+2\,f_{{4}}{a_{{0}}}^{2}a_{{1}}c_{{3}}$ $+2\,f_{{4
}}{a_{{0}}}^{2}a_{{2}}c_{{2}}$ $-4\,f_{{4}}a_{{0}}{a_{{1}}}^{2}c_{{2}}$ $-6\,f_{{4}}a_{{0}}a_
{{1}}a_{{3}}c_{{0}}$ $-2\,f_{{4}}{a_{{1}}}^{3}c_{{1}}$ $-4\,f_{{5}}{a_{{0}}}^{2}a_{{2}}c_{{3}
}$ $+4\,f_{{5}}{a_{{0}}}^{2}a_{{3}}c_{{2}}$ $+2\,f_{{5}}a_{{0}}a_{{1}}a_{{2}}c_{{2}}$ $+4\,f_{{5
}}a_{{0}}a_{{1}}a_{{3}}c_{{1}}$ $+4\,f_{{5}}a_{{0}}{a_{{2}}}^{2}c_{{1}}$ $-2\,f_{{5}}{a_{{1}}
}^{2}a_{{2}}c_{{1}}$ $-8\,f_{{6}}{a_{{0}}}^{2}a_{{3}}c_{{3}}$ $+4\,f_{{6}}a_{{0}}a_{{1}}a_{{3
}}c_{{2}}$ $+4\,f_{{6}}a_{{0}}{a_{{3}}}^{2}c_{{0}}$ $+2\,f_{{7}}a_{{0}}a_{{2}}a_{{3}}c_{{2}}$ $-
12\,f_{{7}}a_{{0}}{a_{{3}}}^{2}c_{{1}}$ $-4\,f_{{7}}{a_{{1}}}^{2}a_{{3}}c_{{2}}$ $-2\,f_{{7}}
a_{{1}}a_{{2}}a_{{3}}c_{{1}}$ $-4\,f_{{7}}a_{{1}}{a_{{3}}}^{2}c_{{0}}
+
8\,f_{{0}}f_{{6}}{a_{{0}}}^{3}a_{{2}}$ $-4\,f_{{0}}f_{{6}}{a_{{0}}}^{2}{a_{{1}}}^{2}$ $+12\,f
_{{0}}f_{{7}}{a_{{0}}}^{3}a_{{3}}$ $+4\,f_{{0}}f_{{7}}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $-4\,f_{{0
}}f_{{7}}a_{{0}}{a_{{1}}}^{3}$ $+{f_{{2}}}^{2}{a_{{0}}}^{4}$ $+2\,f_{{2}}f_{{3}}{a_{{0}}}^{3}
a_{{1}}$ $-2\,f_{{2}}f_{{4}}{a_{{0}}}^{3}a_{{2}}$ $+2\,f_{{2}}f_{{4}}{a_{{0}}}^{2}{a_{{1}}}^{
2}$ $-4\,f_{{2}}f_{{5}}{a_{{0}}}^{3}a_{{3}}$ $+2\,f_{{2}}f_{{5}}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $+4
\,f_{{2}}f_{{6}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $-2\,f_{{2}}f_{{7}}{a_{{0}}}^{2}a_{{2}}a_{{3}
}$ $+4\,f_{{2}}f_{{7}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $+{f_{{3}}}^{2}{a_{{0}}}^{2}{a_{{1}}}^{2}$ $-
2\,f_{{3}}f_{{4}}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $+2\,f_{{3}}f_{{4}}a_{{0}}{a_{{1}}}^{3}$ $-4\,f
_{{3}}f_{{5}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $+2\,f_{{3}}f_{{5}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $+8
\,f_{{3}}f_{{6}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $+12\,f_{{3}}f_{{7}}{a_{{0}}}^{2}{a_{{3}}}^{2
}$ $+2\,f_{{3}}f_{{7}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $+{f_{{4}}}^{2}{a_{{0}}}^{2}{a_{{2}}}^{2}
-2\,{f_{{4}}}^{2}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $+{f_{{4}}}^{2}{a_{{1}}}^{4}$ $+4\,f_{{4}}f_{{5
}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $-8\,f_{{4}}f_{{5}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $+2\,f_{{4}}f_
{{5}}{a_{{1}}}^{3}a_{{2}}$ $+8\,f_{{4}}f_{{6}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $+18\,f_{{4}}f_{{
7}}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $+2\,f_{{4}}f_{{7}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $-2\,f_{{4}}f
_{{7}}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $+4\,{f_{{5}}}^{2}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $+4\,{f_{{5}
}}^{2}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $-4\,{f_{{5}}}^{2}a_{{0}}{a_{{2}}}^{3}$ $+{f_{{5}}}^{2}{a
_{{1}}}^{2}{a_{{2}}}^{2}$ $+8\,f_{{5}}f_{{6}}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $-8\,f_{{5}}f_{{6}}
a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $+4\,f_{{5}}f_{{7}}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $-8\,f_{{5}}f_{{
7}}{a_{{1}}}^{2}{a_{{3}}}^{2}$ $+2\,f_{{5}}f_{{7}}a_{{1}}{a_{{2}}}^{2}a_{{3}}$ $-8\,{f_{{6}}}
^{2}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $-12\,f_{{6}}f_{{7}}a_{{0}}{a_{{3}}}^{3}$ $+8\,f_{{6}}f_{{7}
}a_{{1}}a_{{2}}{a_{{3}}}^{2}$ $+12\,{f_{{7}}}^{2}a_{{1}}{a_{{3}}}^{3}$ $+{f_{{7}}}^{2}{a_{{2}
}}^{2}{a_{{3}}}^{2}$.

\noindent
${\bf eq[26]} =
{a_{{0}}}^{2}c_{{0}}c_{{3}}$ $-{a_{{0}}}^{2}c_{{1}}c_{{2}}$ $-a_{{0}}a_{{1}}{c_{{1}}}^{2}$ $-3\,
a_{{0}}a_{{2}}c_{{0}}c_{{1}}$ $+a_{{0}}a_{{3}}{c_{{0}}}^{2}$ $+{a_{{1}}}^{2}c_{{0}}c_{{1}}
-2\,f_{{1}}{a_{{0}}}^{3}c_{{0}}$ $+f_{{2}}{a_{{0}}}^{3}c_{{1}}$ $-f_{{2}}{a_{{0}}}^{2}a_{{1}}
c_{{0}}$ $+f_{{3}}{a_{{0}}}^{2}a_{{1}}c_{{1}}$ $+2\,f_{{3}}{a_{{0}}}^{2}a_{{2}}c_{{0}}$ $-f_{{3}
}a_{{0}}{a_{{1}}}^{2}c_{{0}}$ $-f_{{4}}{a_{{0}}}^{2}a_{{2}}c_{{1}}$ $-2\,f_{{4}}{a_{{0}}}^{2}
a_{{3}}c_{{0}}$ $+f_{{4}}a_{{0}}{a_{{1}}}^{2}c_{{1}}$ $+3\,f_{{4}}a_{{0}}a_{{1}}a_{{2}}c_{{0}
}$ $-f_{{4}}{a_{{1}}}^{3}c_{{0}}$ $+3\,f_{{5}}{a_{{0}}}^{2}a_{{1}}c_{{3}}$ $+2\,f_{{5}}{a_{{0}}}
^{2}a_{{2}}c_{{2}}$ $-2\,f_{{5}}{a_{{0}}}^{2}a_{{3}}c_{{1}}$ $-4\,f_{{5}}a_{{0}}{a_{{1}}}^{2}
c_{{2}}$ $-5\,f_{{5}}a_{{0}}a_{{1}}a_{{3}}c_{{0}}$ $+3\,f_{{5}}a_{{0}}{a_{{2}}}^{2}c_{{0}}$ $-f_
{{5}}{a_{{1}}}^{3}c_{{1}}$ $-f_{{5}}{a_{{1}}}^{2}a_{{2}}c_{{0}}$ $-2\,f_{{6}}{a_{{0}}}^{2}a_{
{2}}c_{{3}}$ $+2\,f_{{6}}a_{{0}}a_{{1}}a_{{2}}c_{{2}}$ $+2\,f_{{6}}a_{{0}}a_{{1}}a_{{3}}c_{{1
}}$ $+6\,f_{{6}}a_{{0}}{a_{{2}}}^{2}c_{{1}}$ $-2\,f_{{6}}{a_{{1}}}^{2}a_{{2}}c_{{1}}$ $+5\,f_{{7
}}{a_{{0}}}^{2}a_{{3}}c_{{3}}$ $+f_{{7}}a_{{0}}a_{{1}}a_{{2}}c_{{3}}$ $-9\,f_{{7}}a_{{0}}a_{{
1}}a_{{3}}c_{{2}}$ $-f_{{7}}a_{{0}}{a_{{2}}}^{2}c_{{2}}$ $-16\,f_{{7}}a_{{0}}{a_{{3}}}^{2}c_{
{0}}$ $+f_{{7}}{a_{{1}}}^{2}a_{{2}}c_{{2}}$ $-2\,f_{{7}}{a_{{1}}}^{2}a_{{3}}c_{{1}}$ $+2\,f_{{7}
}a_{{1}}a_{{2}}a_{{3}}c_{{0}}
+
4\,f_{{1}}f_{{6}}{a_{{0}}}^{3}a_{{2}}$ $-2\,f_{{1}}f_{{6}}{a_{{0}}}^{2}{a_{{1}}}^{2}$ $+6\,f_
{{1}}f_{{7}}{a_{{0}}}^{3}a_{{3}}$ $+2\,f_{{1}}f_{{7}}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $-2\,f_{{1}
}f_{{7}}a_{{0}}{a_{{1}}}^{3}$ $-2\,f_{{2}}f_{{5}}{a_{{0}}}^{3}a_{{2}}$ $+f_{{2}}f_{{5}}{a_{{0
}}}^{2}{a_{{1}}}^{2}$ $+4\,f_{{2}}f_{{7}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $+f_{{2}}f_{{7}}{a_{{0}
}}^{2}{a_{{2}}}^{2}$ $-2\,f_{{2}}f_{{7}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $-2\,f_{{3}}f_{{5}}{a_{{0
}}}^{2}a_{{1}}a_{{2}}$ $+f_{{3}}f_{{5}}a_{{0}}{a_{{1}}}^{3}$ $-4\,f_{{3}}f_{{6}}{a_{{0}}}^{2}
{a_{{2}}}^{2}$ $+2\,f_{{3}}f_{{6}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $-6\,f_{{3}}f_{{7}}{a_{{0}}}^{
2}a_{{2}}a_{{3}}$ $+4\,f_{{3}}f_{{7}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $-f_{{3}}f_{{7}}a_{{0}}a_{{
1}}{a_{{2}}}^{2}$ $+2\,f_{{4}}f_{{5}}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $-3\,f_{{4}}f_{{5}}a_{{0}}{a
_{{1}}}^{2}a_{{2}}$ $+f_{{4}}f_{{5}}{a_{{1}}}^{4}$ $+4\,f_{{4}}f_{{6}}{a_{{0}}}^{2}a_{{2}}a_{
{3}}$ $-2\,f_{{4}}f_{{6}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $-4\,f_{{4}}f_{{6}}a_{{0}}a_{{1}}{a_{{2
}}}^{2}$ $+2\,f_{{4}}f_{{6}}{a_{{1}}}^{3}a_{{2}}$ $+6\,f_{{4}}f_{{7}}{a_{{0}}}^{2}{a_{{3}}}^{
2}$ $-8\,f_{{4}}f_{{7}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $-f_{{4}}f_{{7}}a_{{0}}{a_{{2}}}^{3}$ $+2\,
f_{{4}}f_{{7}}{a_{{1}}}^{3}a_{{3}}$ $+f_{{4}}f_{{7}}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $+4\,{f_{{5}}
}^{2}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $-8\,{f_{{5}}}^{2}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $+{f_{{5}}}^
{2}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $+{f_{{5}}}^{2}{a_{{1}}}^{3}a_{{2}}$ $+10\,f_{{5}}f_{{6}}a_{{0
}}a_{{1}}a_{{2}}a_{{3}}$ $-6\,f_{{5}}f_{{6}}a_{{0}}{a_{{2}}}^{3}$ $+2\,f_{{5}}f_{{6}}{a_{{1}}
}^{2}{a_{{2}}}^{2}$ $-3\,f_{{5}}f_{{7}}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $-f_{{5}}f_{{7}}a_{{0}}{a
_{{2}}}^{2}a_{{3}}$ $+3\,f_{{5}}f_{{7}}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $-4\,{f_{{6}}}^{2}a_{{0}}
{a_{{2}}}^{2}a_{{3}}$ $+20\,f_{{6}}f_{{7}}a_{{0}}a_{{2}}{a_{{3}}}^{2}$ $-4\,f_{{6}}f_{{7}}a_{
{1}}{a_{{2}}}^{2}a_{{3}}$ $+39\,{f_{{7}}}^{2}a_{{0}}{a_{{3}}}^{3}$ $-5\,{f_{{7}}}^{2}a_{{1}}a
_{{2}}{a_{{3}}}^{2}$ $-{f_{{7}}}^{2}{a_{{2}}}^{3}a_{{3}}$.

\noindent
${\bf eq[27]} = 
{a_{{0}}}^{2}c_{{1}}c_{{3}}$ $-3\,a_{{0}}a_{{1}}c_{{1}}c_{{2}}$ $+2\,a_{{0}}a_{{2}}c_{{0}}c_{
{2}}$ $-10\,a_{{0}}a_{{3}}c_{{0}}c_{{1}}$ $-2\,{a_{{1}}}^{2}{c_{{1}}}^{2}$ $+a_{{1}}a_{{2}}c_{{0
}}c_{{1}}$ $+a_{{1}}a_{{3}}{c_{{0}}}^{2}
+
8\,f_{{0}}{a_{{0}}}^{3}c_{{0}}$ $+f_{{1}}{a_{{0}}}^{3}c_{{1}}$ $-f_{{1}}{a_{{0}}}^{2}a_{{1}}c
_{{0}}$ $+2\,f_{{2}}{a_{{0}}}^{3}c_{{2}}$ $+4\,f_{{2}}{a_{{0}}}^{2}a_{{1}}c_{{1}}$ $-2\,f_{{2}}{
a_{{0}}}^{2}a_{{2}}c_{{0}}$ $+2\,f_{{3}}{a_{{0}}}^{2}a_{{1}}c_{{2}}$ $-f_{{3}}{a_{{0}}}^{2}a_
{{2}}c_{{1}}$ $+8\,f_{{3}}{a_{{0}}}^{2}a_{{3}}c_{{0}}$ $+4\,f_{{3}}a_{{0}}{a_{{1}}}^{2}c_{{1}
}$ $-f_{{3}}a_{{0}}a_{{1}}a_{{2}}c_{{0}}$ $-5\,f_{{4}}{a_{{0}}}^{2}a_{{1}}c_{{3}}$ $-2\,f_{{4}}{
a_{{0}}}^{2}a_{{2}}c_{{2}}$ $+f_{{4}}{a_{{0}}}^{2}a_{{3}}c_{{1}}$ $+7\,f_{{4}}a_{{0}}{a_{{1}}
}^{2}c_{{2}}$ $+12\,f_{{4}}a_{{0}}a_{{1}}a_{{3}}c_{{0}}$ $+2\,f_{{4}}a_{{0}}{a_{{2}}}^{2}c_{{0
}}$ $+4\,f_{{4}}{a_{{1}}}^{3}c_{{1}}$ $-f_{{4}}{a_{{1}}}^{2}a_{{2}}c_{{0}}$ $+14\,f_{{5}}{a_{{0}
}}^{2}a_{{2}}c_{{3}}$ $+2\,f_{{5}}{a_{{0}}}^{2}a_{{3}}c_{{2}}$ $-14\,f_{{5}}a_{{0}}a_{{1}}a_{
{2}}c_{{2}}$ $-3\,f_{{5}}a_{{0}}a_{{1}}a_{{3}}c_{{1}}$ $-16\,f_{{5}}a_{{0}}{a_{{2}}}^{2}c_{{1
}}$ $+3\,f_{{5}}{a_{{1}}}^{2}a_{{2}}c_{{1}}$ $-2\,f_{{5}}{a_{{1}}}^{2}a_{{3}}c_{{0}}$ $-f_{{5}}a
_{{1}}{a_{{2}}}^{2}c_{{0}}$ $+18\,f_{{6}}{a_{{0}}}^{2}a_{{3}}c_{{3}}$ $-2\,f_{{6}}a_{{0}}a_{{
1}}a_{{2}}c_{{3}}$ $-14\,f_{{6}}a_{{0}}a_{{1}}a_{{3}}c_{{2}}$ $-2\,f_{{6}}a_{{0}}{a_{{2}}}^{2
}c_{{2}}$ $-10\,f_{{6}}a_{{0}}{a_{{3}}}^{2}c_{{0}}$ $+2\,f_{{6}}{a_{{1}}}^{2}a_{{2}}c_{{2}}$ $-2
\,f_{{6}}a_{{1}}a_{{2}}a_{{3}}c_{{0}}$ $+2\,f_{{7}}a_{{0}}{a_{{2}}}^{2}c_{{3}}$ $-10\,f_{{7}}
a_{{0}}a_{{2}}a_{{3}}c_{{2}}$ $+31\,f_{{7}}a_{{0}}{a_{{3}}}^{2}c_{{1}}$ $+4\,f_{{7}}{a_{{1}}}
^{2}a_{{3}}c_{{2}}$ $-2\,f_{{7}}a_{{1}}a_{{2}}a_{{3}}c_{{1}}$ $-f_{{7}}{a_{{2}}}^{3}c_{{1}}$ $-f
_{{7}}{a_{{2}}}^{2}a_{{3}}c_{{0}}
-16\,f_{{0}}f_{{6}}{a_{{0}}}^{3}a_{{2}}$ $+8\,f_{{0}}f_{{6}}{a_{{0}}}^{2}{a_{{1}}}^{2}$ $-24
\,f_{{0}}f_{{7}}{a_{{0}}}^{3}a_{{3}}$ $-8\,f_{{0}}f_{{7}}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $+8\,f_
{{0}}f_{{7}}a_{{0}}{a_{{1}}}^{3}$ $-2\,f_{{1}}f_{{5}}{a_{{0}}}^{3}a_{{2}}$ $+f_{{1}}f_{{5}}{a
_{{0}}}^{2}{a_{{1}}}^{2}$ $+4\,f_{{1}}f_{{7}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $+f_{{1}}f_{{7}}{a_
{{0}}}^{2}{a_{{2}}}^{2}$ $-2\,f_{{1}}f_{{7}}a_{{0}}{a_{{1}}}^{2}a_{{2}}$ $-2\,{f_{{2}}}^{2}{a
_{{0}}}^{4}$ $-4\,f_{{2}}f_{{3}}{a_{{0}}}^{3}a_{{1}}$ $+4\,f_{{2}}f_{{4}}{a_{{0}}}^{3}a_{{2}}
-4\,f_{{2}}f_{{4}}{a_{{0}}}^{2}{a_{{1}}}^{2}$ $+2\,f_{{2}}f_{{5}}{a_{{0}}}^{3}a_{{3}}$ $-2\,f
_{{2}}f_{{5}}{a_{{0}}}^{2}a_{{1}}a_{{2}}$ $-8\,f_{{2}}f_{{6}}{a_{{0}}}^{2}a_{{1}}a_{{3}}$ $+2
\,f_{{2}}f_{{6}}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $+12\,f_{{2}}f_{{7}}{a_{{0}}}^{2}a_{{2}}a_{{3}
}$ $-8\,f_{{2}}f_{{7}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $-2\,f_{{2}}f_{{7}}a_{{0}}a_{{1}}{a_{{2}}}
^{2}$ $-2\,{f_{{3}}}^{2}{a_{{0}}}^{2}{a_{{1}}}^{2}$ $+4\,f_{{3}}f_{{4}}{a_{{0}}}^{2}a_{{1}}a_
{{2}}$ $-4\,f_{{3}}f_{{4}}a_{{0}}{a_{{1}}}^{3}$ $+2\,f_{{3}}f_{{5}}{a_{{0}}}^{2}a_{{1}}a_{{3}
}$ $+2\,f_{{3}}f_{{5}}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $-3\,f_{{3}}f_{{5}}a_{{0}}{a_{{1}}}^{2}a_{{
2}}$ $-16\,f_{{3}}f_{{6}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $+2\,f_{{3}}f_{{6}}a_{{0}}a_{{1}}{a_{{2
}}}^{2}$ $-24\,f_{{3}}f_{{7}}{a_{{0}}}^{2}{a_{{3}}}^{2}$ $-f_{{3}}f_{{7}}a_{{0}}{a_{{2}}}^{3}
-2\,{f_{{4}}}^{2}{a_{{0}}}^{2}{a_{{2}}}^{2}$ $+4\,{f_{{4}}}^{2}a_{{0}}{a_{{1}}}^{2}a_{{2}}
-2\,{f_{{4}}}^{2}{a_{{1}}}^{4}$ $-4\,f_{{4}}f_{{5}}{a_{{0}}}^{2}a_{{2}}a_{{3}}$ $+13\,f_{{4}}
f_{{5}}a_{{0}}{a_{{1}}}^{2}a_{{3}}$ $-f_{{4}}f_{{5}}a_{{0}}a_{{1}}{a_{{2}}}^{2}$ $-3\,f_{{4}}
f_{{5}}{a_{{1}}}^{3}a_{{2}}$ $-18\,f_{{4}}f_{{6}}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $-2\,f_{{4}}f_
{{6}}a_{{0}}{a_{{2}}}^{3}$ $+2\,f_{{4}}f_{{6}}{a_{{1}}}^{2}{a_{{2}}}^{2}$ $-35\,f_{{4}}f_{{7}
}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $-11\,f_{{4}}f_{{7}}a_{{0}}{a_{{2}}}^{2}a_{{3}}$ $+6\,f_{{4}}f_
{{7}}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $+f_{{4}}f_{{7}}a_{{1}}{a_{{2}}}^{3}$ $+4\,{f_{{5}}}^{2}{a_
{{0}}}^{2}{a_{{3}}}^{2}$ $-32\,{f_{{5}}}^{2}a_{{0}}a_{{1}}a_{{2}}a_{{3}}$ $+16\,{f_{{5}}}^{2}
a_{{0}}{a_{{2}}}^{3}$ $+{f_{{5}}}^{2}{a_{{1}}}^{3}a_{{3}}$ $-{f_{{5}}}^{2}{a_{{1}}}^{2}{a_{{2
}}}^{2}$ $-32\,f_{{5}}f_{{6}}a_{{0}}a_{{1}}{a_{{3}}}^{2}$ $+26\,f_{{5}}f_{{6}}a_{{0}}{a_{{2}}
}^{2}a_{{3}}$ $+8\,f_{{5}}f_{{6}}{a_{{1}}}^{2}a_{{2}}a_{{3}}$ $-8\,f_{{5}}f_{{7}}a_{{0}}a_{{2
}}{a_{{3}}}^{2}$ $+12\,f_{{5}}f_{{7}}{a_{{1}}}^{2}{a_{{3}}}^{2}$ $+3\,f_{{5}}f_{{7}}a_{{1}}{a
_{{2}}}^{2}a_{{3}}$ $+f_{{5}}f_{{7}}{a_{{2}}}^{4}$ $+20\,{f_{{6}}}^{2}a_{{0}}a_{{2}}{a_{{3}}}
^{2}$ $+30\,f_{{6}}f_{{7}}a_{{0}}{a_{{3}}}^{3}$ $-6\,f_{{6}}f_{{7}}a_{{1}}a_{{2}}{a_{{3}}}^{2
}$ $+2\,f_{{6}}f_{{7}}{a_{{2}}}^{3}a_{{3}}$ $-8\,{f_{{7}}}^{2}a_{{1}}{a_{{3}}}^{3}$.

}

Subject to Conjecture~\ref{conj:kum}, and by analogy with the defining
equation of the Kummer surface in genus~2, which is only quadratic
in~$k_3$, we make the following conjecture.

\begin{conj}\label{conj:kummdef}
The equations of Theorem~\ref{thm:kummdefg3} are a set of defining
equations for the Kummer variety in genus~3.
\end{conj}

It would be nice to be able to check all quartic monomials, including
those quartic in $c_0,c_1,c_2,c_3$. However, these are beyond
storage capacity. Note that even to verify that the above
are identities cannot be done all at once; one needs to slice
each of the above into portions and use intermediary simplification
steps.

Note that one consequence of the equation $eq[1]=0$, above, is that
there is a unique linear relation between the quadratic monomials
in the $\{ \ka_i \} = \{ a_i\} \cup \{ c_i\}$ of Theorem~\ref{thm:g3kum}.
This has a surprising consequence for the even functions
of $L(4\Theta)$, which does not happen in genus~1 or~2.

\begin{theorem}\label{thm:evenshort}
The $(4^g + 2^g)/2 = 36$ quadratic monomials in the $\ka_i$ of Theorem~\ref{thm:g3kum}
span a $((4^g + 2^g)/2 - 1) = 35$-dimensional subspace
of the $(4^g + 2^g)/2 = 36$-dimensional space of even functions
of $L(4\Theta)$.
\end{theorem}

\begin{proof}
Our method of finding all of the equations in Theorem~\ref{thm:kummdefg3}
included finding all possible quadratic relations on $L(2\Theta)$,
of which there is precisely one. There are, in total,
$(4^g + 2^g)/2$ quadratic monomials in the $\ka_i$ of Theorem~\ref{thm:g3kum},
all of which are even functions in $L(4\Theta)$, and
which span one dimension less, due to the above linear dependency.
We know that the set of even functions of $L(4\Theta)$
has dimension $(4^g + 2^g)/2$ by Lemma~\ref{lem:oddeven}.
\end{proof}

Contrast this with genus~1 and~2. In any genus, there are $(4^g + 2^g)/2$ 
quadratic monomials in the coordinate functions of $L(2\Theta)$,
and the space of even functions of $L(4\Theta)$ is also $(4^g + 2^g)/2$.
In genus~1 and~2, the quadratic monomials in the coordinate functions of $L(2\Theta)$
are linearly independent, and so give a basis for the even
functions of $L(4\Theta)$. For example, in genus~1,
the $\ka_0 = 1,\ \ka_1 = x$ of~(\ref{eq:ellkumrhoz}) give
the three quadratic monomials $\ka_0^2 , \ka_0 \ka_1, \ka_1^2$,
which give a basis for the span of the three even functions $z_0, z_1, z_3$
of~(\ref{eq:ellkumrhoz}). Similarly in genus~2 the
$\ka_0 ,\ka_1, \ka_2, \ka_3$ of~(\ref{eq:gen2kum})
give the 10~quadratic monomials $\ka_0^2, \ka_0 \ka_1, \ka_0 \ka_2,
\ka_0 \ka_3, \ka_1^2, \ka_1 \ka_2, \ka_1 \ka_3, \ka_2^2, \ka_2 \ka_3, \ka_3^2$
which span the same space as the 10~even function
of $L(4\Theta)$ given by $z_0,z_3,z_4,z_5$, $z_{10},z_{11},z_{12},
z_{13},z_{14},z_{15}$ of~(\ref{eq:gen2z}).  
Hence, genus~3 is the first genus for which there is
still an even function to be found in $L(4\Theta)$
beyond the quadratic monomials of the basis for $L(2\Theta)$.

\section{Relations in $L(4\Theta)$ for Genus~$3$.}\label{sec:defL34T}

Since we have not given a complete basis for $L(3\Theta)$
or $L(4\Theta)$ we cannot of course give a complete set of
defining equations in either case.
Theorem~\ref{thm:g3L3} describes what portion of $L(3\Theta)$ we know.
For $L(4\Theta)$, we have the 35 even functions $e_{ij}$ of
Theorem~\ref{thm:evenshort}, given by the monomials

\begin{equation}\label{eq:eij}
e_{ij} = \ka_i \ka_j, \; \; 0 \le i \le j \le 7, \; (i,j) \not= (0,7).
\end{equation}

We have excluded $(i,j) = (0,7)$ since $\ka_0 \ka_7$,
which is the same as $a_0 c_3$, is linearly dependent on
the other $e_{ij}$ by Theorem~\ref{thm:kummdefg3}.
There are also the $b_0,b_1,b_2$ of~(\ref{eq:gen3abd}),  
giving a total of~$38$ independent members of
the $4^g = 64$-dimensional space $L(4\Theta)$.
Twelve of the missing members of the basis for $L(4\Theta)$
are the missing members of the basis for $L(3\Theta)$.
Recall from Lemma~\ref{lem:ample} that either $L(3\Theta)$
or $L(4\Theta)$ yields an embedding of the Jacobian
variety, with the $L(3\Theta)$ embedding in $\PP^{26}$ having defining
equations at most cubic, and the $L(4\Theta)$ embedding
in $\PP^{63}$ having quadratic defining equations, as 
a special case of a theorem of Mumford
(\cite{mumford:abelian},~I), which states that the Jacobian
can be expressed as an intersection  of quadratics.

Many of these are easy to find. First of all, there
are the curve-independent equations:

\begin{equation}\label{eq:curindep}
e_{ij} e_{k\ell} = e_{ik} e_{j\ell}, \ \ i,j,k,\ell \in \{ 0,\ldots ,7\},\ j\not= k.
\end{equation}  

These are analogous to the equation
$z_1^2 = z_0 z_3$ in~(\ref{eq:ellz})
for the $\PP^3$ embedding $z_0,\ldots ,z_3$
of an elliptic curve given in~(\ref{eq:ellkumrhoz}),
and to equations of the form $z_0 z_{14} = z_5^2$
for the $\PP^{15}$ embedding of the Jacobian
of a curve of genus~2 described in~(\ref{eq:gen2z}).
In addition, every one of the quartic equations
in Theorem~\ref{thm:kummdefg3} induces a quadratic
form on the $e_{ij}$ obtained by replacing quadratic
monomials $\ka_i \ka_j$ by $e_{ij}$, and by further
using ${\bf eq[1]}$ to eliminate any occurrence of $\ka_0 \ka_7$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \input future.tex

\chapter{Future Applications}
\label{ch:future}

Much of this section will describe what should
be true, with outlines of a strategy rather than proofs. 

An explicit embedding of the Jacobian has potential applications to
Chabauty's Theorem (\cite{chab:ratpoints} and Theorem~13.1.2 in
Cassels and Flynn~\cite{prolegom}), which states that the number of
rational points on a curve over a number field is finite if the rank
of the Jacobian is less than the genus of the curve.  One approach to
implementing Chabauty's Theorem in practice has been to derive
explicit local power series expansions from the formal group. In genus
2, it has proved useful to have an explicit embedding of the Jacobian
variety, from which the terms of the formal group can be derived, and
there are now many genus 2 applications of Chabauty's Theorem in the
literature (\cite{bruin:238}, \cite{chab:ratpoints}, \cite{coleman},
\cite{flynn:flexchab}, \cite{fps}, \cite{fw1}, \cite{flynn:qder},
\cite{fw}, and \cite{mccallum}).

We shall concentrate on the potential of our results eventually to be
applied to give the explicit computation of height constants in
genus~3. Much work has already been done in this area for genus~1 and
genus~2, which gives rise to applications involving Selmer groups and
descent techniques (\cite{dja}, \cite{fs}, \cite{sch:selfun},
\cite{sik}, \cite{stoll:ht1}, \cite{stoll:twodesc}, and
\cite{stoll:ht2}).

We first briefly 
summarise how these height constants can be computed in genus~1 and~2.
We first recall the standard definition of a height function
on an Abelian group.

\begin{define}\label{def:heightfn}
Let $G$ be an abelian group. A
{\em height function} $H$ on $G$ is a map $H : G \mapsto {\bbbr}^+$ satisfying:
\par
{\obeylines
\noindent{(1).} There exists a constant $C_1$ such that, for all $P, Q\in G$,
\ \ \ \ $H(P+Q)H(P-Q)\leqslant C_1 H(P) ^2 H(Q)^2$.
\noindent {(2).} There exists a constant $C_2$ such that, for all $P\in G$, $H(2P)\geqslant H(P) ^4/C_2$.
\noindent {(3).} For any constant $C_3$, the set $\{ P\in G : H(P) \leqslant C_3\}$ is finite.
}
\par\noindent The constants $C_1$, $C_2$ depend only on the group $G$
and the height function $H$,
and we shall refer to them as the {\em height constants}. 
\end{define}

The following property of abelian groups with a height function
is proved in \cite{sil:AEC1}, p.199.

\begin{lemma}\label{lem:heights}
Let $G$ be an abelian group with height function
$H$, such that $G/2G$ is a finite set: $\{ Q_1,\ldots ,Q_n\}$, say.
Then $G$ is finitely generated. Explicitly, if $\epsilon = 
\hbox{min}\{ H(P) : P \in G \}$, and $C_1' =
\hbox{max}\{ H(Q_i)^2 : 1\leqslant i \leqslant n\} \cdot C_1 / \epsilon$,
then $G$ is generated by the finite set: $\{ P\in G : H(P) \leqslant
\sqrt{C_1' C_2}\} \cup \{ Q_1,\ldots ,Q_n\}$.
\end{lemma}

When $K$ is a number field,
there is a standard function $H$ which may be defined on $\PP^n(K)$.

\begin{define}\label{def:htpnk}
Let ${\bf x} = 
(x_i) \in \PP^n(K)$.
Define $H({\bf x}) = \bigl( 
\prod\limits_v
\mathop{\hbox{max}}
\limits_{0\leqslant i\leqslant n} \vert x_i\vert _v \bigr)
^{1/[K:{\bbbq}]}$, where
$\prod\limits_v$ is over all valuations
on $K$ which extend the usual valuations $\vert \ \vert _p$
and $\vert \ \vert _\infty$ on $\bbbq$.
%For any $x\in K$, define $H(x) = H((\frac{1}{x}))$.
\end{define}

When $K = \bbbq$ this is the same as first representing
{\bf x} by $(x_i)$, where $x_i \in \bbbz$ and
$\hbox{gcd}(x_1,\ldots ,x_n) = 1$, and defining
$H({\bf x}) = \mathop{\hbox{max}}\limits_{0\leqslant i\leqslant n} \vert x_i\vert$.
The following is a standard fact about heights (Theorem 5.11
in \cite{sil:AEC1}).

\begin{lemma}\label{lem:finiteh}
For any $C$, the set $\{ {\bf x} \in {\PP}^n(K) :
H({\bf x}) \leqslant C\}$ is finite.
\end{lemma}

For an elliptic curve and the Jacobian of a curve of genus~2,
one can define a function using the $L(2\Theta)$ embedding
of the Kummer variety.

\begin{define}\label{def:kumhtg1g2} 
Let $\E$ be an elliptic curve, as in~(\ref{eq:ell}).
Define the {\em Kummer height} $H_\ka$ on~$\E$ by
$H_\ka (P) = H(\ka_0(P), \ka_1(P))$, where the~$\ka_i$
are as in~(\ref{eq:ellkumrhoz}).
Let~$\C$
be a curve of genus~$2$, as in~(\ref{eq:genus2}), with Jacobian~$J$. 
Define the {\em Kummer height} $H_\ka$ on~$J(K)$ by
$H_\ka (D) = H(\ka_0(D), \ldots ,\ka_3(D))$, where the~$\ka_i$
are as in~(\ref{eq:gen2kum}).
\end{define}

It is immediate from Lemma~\ref{lem:finiteh} that
$H_\ka$ always satisfies axiom~(3) of Definition~\ref{def:heightfn}. 
For elliptic curves, this height is sometimes 
called the {\em $x$-coordinate height}.
In both genus~1 and~2,
there are polynomials $B_{ij}$
biquadratic in the $\ka_j(P)$, $\ka_j(Q)$ and defined
over~$K$ such that

\begin{equation}\label{eq:bij}
\bigl( \ka_i(P+Q)\ka_j(P-Q)
+\ka_i(P-Q)\ka_j(P+Q)\bigr) =2\bigl( B_{ij}(P,Q)\bigr).
\end{equation}

For elliptic curves these are given on
p.416 of~\cite{sil:AEC1}.
In the genus~2 case, see p.23 of~\cite{prolegom}.
These biquadratic forms easily give a constant~$C_1$,
dependent only on the coefficients of the~$B_{ij}$,
such that axiom~(1) of Definition~\ref{def:heightfn}
is satisfied. Furthermore, setting $A=B$ gives the
$\bigl( \ka_i(2A) \bigr)$ in terms of non-degenerate
quartics. Combining this with Hilbert's Nullstellensatz
gives a constant~$C_2$ for which axiom~(2) of Definition~\ref{def:heightfn} 
is satisfied. We may therefore deduce the following.

\begin{theorem}\label{thm:htg1g2}
Let $K$, $\E$, $\C$, $J$, $H_\ka$ be as in Definition~\ref{def:kumhtg1g2}.
Then the $H_\ka$ defined on~$\E(K)$ in Definition~\ref{def:kumhtg1g2}
is a height function. Similarly, the
$H_\ka$ defined on~$J(K)$ in Definition~\ref{def:kumhtg1g2} 
is also a height function.
\end{theorem}

It follows that, for elliptic curves and Jacobians of
curves in genus~2 we have an effective procedure for
going from a description of $\E (K) /2\E(K)$ or $J(K) /2J(K)$
(as can sometimes be obtained by a 2-descent)
to a set of generators for the Mordell-Weil group
$\E(K)$ or $J(K)$, respectively. Furthermore, the
height constant~$C_2$ can be improved by using additional
structure on the duplication law. One first notes that
addition by a point of order~2 induces a linear
map on $(\ka_i)$, both in genus~1 and~2 (see Chapter~12 of~\cite{prolegom}).
Choosing a set of $2^g$ points of order~2 as the kernel
of a $2^g$-isogeny, one can simultaneously diagonalise
the corresponding linear maps. If one does this also
to the dual isogeny, it is possible to express
the duplication map on the Kummer as

\begin{equation}\label{eq:kumdup}
[2] = M_1\tau M_2\tau M_3,
\end{equation}

\noindent
where $M_1,M_2,M_3$ are matrices defined over some
field extension~$L$, and $\tau : (\ka_i) \mapsto (\ka_i^2)$.
A height constant $C_2$ can then be derived from the
entries of these matrices, which improves on the height
constant derived directly from applying Hilbert's Nullstellensatz
to the quartic forms.

It is natural to ask whether the same can be attempted in
genus~3, using the embedding we have found for the
Kummer variety.

\begin{define}\label{def:kumhtg3}
Let $\C$ be a hyperelliptic curve of genus~3,
as in~(\ref{eq:hyper}) with $n=7$, with Jacobian~$J$.
Define the {\em Kummer height} $H_\ka$ on~$J(K)$ by
$H_\ka (D) = H(\ka_0(D), \ldots ,\ka_7(D))$, where the~$\ka_i$
are as in Theorem~\ref{thm:g3kum}.
\end{define}

By analogy with the genus~1 and~2 situations, it may be possible
to examine the morphism $\ka^{(2)} \rightarrow \ka^{(2)}$
(where $\ka^{(2)}$ is the
symmetric square of the Kummer variety) that is given by mapping the
unordered pair $\{\pm P, \pm Q\}$ to $\{\pm(P+Q), \pm(P-Q)\}$, and
deduce biquadratic forms analogous to those of~(\ref{eq:bij}),
and so non-degenerate quartic forms for the duplication law.
Furthermore, Lemma~2,4,7 and Proposition~6.1.1 of~\cite{LB}
suggest that addition by a point of order~2 should induce
a linear map on the Kummer variety. If there are such
objects, then the following would be true, by imitating 
the arguments for genus~1 and~2.

\begin{conj}\label{conj:htg3}
Let $\C$, $J$, $H_\ka$ be as in Definition~\ref{def:kumhtg3}.
Then the $H_\ka$ defined on~$J(K)$ in Definition~\ref{def:kumhtg3}
is a height function.
\end{conj}

It would be of interest to find explicitly the corresponding
height constants $C_1,C_2$ since these would allow (for the first time) the actual
computation of generators for~$J(K)$ for those examples where
$J(K)/2J(K)$ has already been found. A further benefit of
finding $C_2$ explicitly is the following immediate consequence of
axiom~(2). 

\begin{corollary}\label{cor:httors}
Let $H$ be a height function on an Abelian group~$A$.
Any $P \in A$ of finite order satisfies
$H(A) \leq C_2^{1/3}$.
\end{corollary}

In view of axiom~(3), it follows that any Abelian group
with a height function has a finite torsion subgroup.
If, in our case, we could find in genus~3 an explicit
expression for~$C_2$ in terms of the coefficients of
the curve~$\C$ (as already exists in genus~1 and~2)
then we would automatically have an effective procedure
for computing the torsion subgroup of~$J(K)$.

A further target for the future would be to understand
how to embed the Kummer variety for genus~3 curves of
the form $y^2 = F(x)$, where $F(x)$ has degree~8.
As already remarked in Section~\ref{sec:L2Tgen3},
the problem is in understanding the representation of divisor classes
by sets of points, using maps only defined over the ground field.
If we set aside this problem, and formally
apply our usual algorithm to $y^2 = f_8 x^8 + \ldots + f_0$,
we obtain variants of the $c_i$ in~(\ref{eq:g3kum}). 

\begin{equation}\label{eq:g3kumvar}
\begin{align*}
c_0' &= c_0 + f_8(2 a_2 a_1^2 - a_1^4 - a_2^2),\\
c_1' &= c_1 + f_8(4 a_2 a_1^3 - a_1^5 -3 a_2^2 a_1 - 2 a_3 a_1^2 +2 a_3 a_2^3),\\
c_2' &= c_2 + f_8(a_2 a_1^4 - 3 a_2^2 a_1^2 + a_2^3 + 2 a_3 a_2 a_1 - a_3^2),\\
c_3' &= c_3 + f_8(a_2^2 a_1^3 - a_3 a_1^4 - 2 a_2^3 a_1
                 + 2 a_3 a_2 a_1^2 + a_3 a_2^2 - a_3^2 a_1).\\
\end{align*}
\end{equation}

What is interesting is that these still have $\wt$ equal to~$2$, hinting
at the possibility of describing the Kummer variety even in this case.
It would also be interesting to try to describe the Kummer variety
for the case where a genus~3 curve is a smooth plane quartic
(every genus~3 curve is either hyperelliptic or a smooth plane quartic),
but we have no idea how to do this.

For hyperelliptic curves of genus~4, there are no problems
with representing divisor classes
by sets of points, using maps only defined over the ground field
since, as in genus~2, we have available $(g/2)\Theta^+ + (g/2)\Theta^-$. 
In principle, our ideas should be able to describe the Kummer
variety for both the cases where $F(x)$ has degree~9 and degree~10.
In particular, it is straightforward to check that the reduction
of $b_0^2$ modulo $\delta^2$ gives us a new basis member.
The difficulty in genus~4 is primarily the size of the expressions
involved and the number of monomials to be checked.

\section{Addition on the Jacobian of a curve of genus $g$}

Cassels~\cite{cas:lectures} recites well-known formul\ae{} for
explicitly expressing the addition of points on the Jacobian of an
elliptic curve in terms of points on the original curve; Cassels and
Flynn~\cite{prolegom} give an equivalent method in Chapter 8 for
curves of genus $2$. Generalizing the approach may be a useful tool
for future work.

Given the curve $\curve{C}:y^2=F(x)$, with $\partial F=2g+1$ and the
divisor $D=\sum m_i P_i$ s.t. $P_i=(x_i,y_i)$, $x_i=x_j\text{ iff }i=j$,
and define $d=2g-\deg D\geq 0$; we seek
the curve $\curve{D}:q(x)y = p(x)$,
$\partial q\leq\floor{\frac{g-d-1}{2}}$ 
and $\partial p\leq\floor{\frac{3g-d}{2}}$,
s.t., for each $P_i$ and $0\leq n < m_i$,
\begin{equation}
\frac{d^n}{dx^n}y_{\curve{C}}\bigg|_{P_i}
 =\frac{d^n}{dx^n}y_{\curve{D}}\bigg|_{P_i}\text{.}
\end{equation}

Note that the L.H.S. of the above can be expressed purely in terms of
$y_i$ and the values of the first $n$ derivatives of $F$ evaluated at
$x_i$, which are relatively easy to calculate without recourse to
methods of symbolic computation.

We find ourselves in a situation
with $\deg D=2g-d$ equations linear in
$\floor{\frac{g-d-1}{2}}+\floor{\frac{3g-d}{2}}+1=2g-d$ unknowns (the
coefficients of $x^k$ in $p$ and $q$).  The curve $\curve{D}$ can then
be recovered using simple linear algebra. In fact, we can simplify the
calculation still further by removing the dependency on using a
symbolic maths package to calculate the derivatives of $y^2$ wrt $x$.


\begin{define}
Define the rational polynomial $h_k$ in $k+1$ variables $d_0,\ldots,d_k$
as follows:
\begin{equation}
h_k(d_0,\ldots,d_k)=
\begin{cases}
d_0              & k=0\\
\frac{d_1}{2d_0} & k=1\\
\frac{1}{2d_0}\bigg(d_k-\sum_{j=1}^{k-1}\binom{k}{j}h_j h_{k-j}\bigg) & k\geq 2
\end{cases}
\end{equation}
\end{define}

\begin{theorem}\label{thm:hkdiff}
Given a function $P(X)$, $n$-times differentiable, then 
\[\sderiv[k]{P}(x)=h_k(d_0,d_1,d_2,\ldots,d_k)\]
where $0\leq k\leq n$, $d_0=P(x)$, and $d_i$ is the $i$th derivative
of $P(X)^2$ evaluated at $x$.
\end{theorem}

\begin{proof}
The result is obviously true for $k=1$, so let us assume that it is true
for all $k<r<n$.
It can be shown that for all $s\leq n$
\[
	\deriv[s]{X}{}\bigg[P(X)^2\bigg]=
		\sum_{i=0}^{s}\binom{s}{i}\sderiv[i]{P}(X)\sderiv[s-i]{P}(X)
\]
Hence
\begin{align*}
\sderiv[r]{P}(x) & =\frac{1}{2d_0}\bigg(
	\deriv[r]{X}{}\bigg[P(X)^2\bigg]_{X=x}
		- \sum_{j=1}^{r-1}\binom{r}{j}
			\sderiv[j]{P}(x)\sderiv[r-j]{P}(x)\bigg)\\
 & = \frac{1}{2d_0}\bigg(d_r
		-\sum_{j=1}^{r-1}\binom{r}{j}h_j h_{r-j}\bigg)
\end{align*}
as required.
\end{proof}

\begin{define}
Given a list of $n+1$ ordered pairs $(x_0,m_0),\ldots,(x_n,m_n)$,
define $N=\sum m_i$. Then take the $N$ monomials $1,x,x^2,\ldots,x^{N-1}$
and define the $N\times N$ matrix $\exvdm \big(\{(x_i,m_i)\}\big)$ by placing
in the first row and $i$th column the value of the monomial $x^{i-1}$ at
$x_0$, into the second row and $i$th column the value of the first
derivative of the monomial $x^{i-1}$ at $x_0$, and so on up until the
$m_0$th row wherein the value of the $(m_0-1)$th derivative of $x^{i-1}$
at $x=x_0$ is placed into the $i$th column. Repeat this process 
in the following rows for each of the $x_i$.
\end{define}

\begin{example}
\[
\exvdm(\{(x_0,3),(x_1,4),(x_2,1)\})=
\begin{bmatrix}
1 & x_0 & x_0^2 & x_0^3  & x_0^4   & x_0^5   & x_0^6    & x_0^7\\
0 & 1   & 2x_0  & 3x_0^2 & 4x_0^3  & 5x_0^4  & 6x_0^5   & 7x_0^6\\
0 & 0   & 2     & 6x_0   & 12x_0^2 & 20x_0^3 & 30x_0^4  & 42x_0^5\\
1 & x_1 & x_1^2 & x_1^3  & x_1^4   & x_1^5   & x_1^6    & x_1^7\\
0 & 1   & 2x_1  & 3x_1^2 & 4x_1^3  & 5x_1^4  & 6x_1^5   & 7x_1^6\\
0 & 0   & 2     & 6x_1   & 12x_1^2 & 20x_1^3 & 30x_1^4  & 42x_1^5\\
0 & 0   & 0     & 6      & 24x_1   & 60x_1^2 & 120x_1^3 & 210x_1^4\\
1 & x_2 & x_2^2 & x_2^3  & x_2^4   & x_2^5   & x_2^6    & x_2^7
\end{bmatrix}
\]
\end{example}

\begin{theorem}\label{thm:exvdm}
\[
\det \exvdm(\{(x_i,m_i)\}) =
	\prod_{0\leq i < j \leq n}(x_j-x_i)^{m_i m_j}
		\cdot\prod_{i=0}^{n}\prod_{k=0}^{m_i-1}k!
\]
\end{theorem}

\begin{proof}
See Proposition 4.2.1 in \cite{vvg:1}.
\end{proof}

Note that in the case where $m_i=1$ for every $i$, we have the
usual Vandermonde determinant of the $x_i$'s.

\begin{example}
\[
\det \exvdm(\{(x_0,3),(x_1,4),(x_2,1)\})=
	24(x_2-x_1)^4(x_1-x_0)^{12}(x_2-x_0)^3
\]
\end{example}

\begin{theorem}
Given a hyperelliptic curve $\mathcal{C}:Y^2=F(X)$
and a list $(x_i,y_i)$ of $n$ distinct points each with multiplicity $m_i$
s.t. $x_i\neq x_j$ whenever $i\neq j$. Suppose that $d=\big(\sum
m_i\big)-1$. Then
there is a unique polynomial
$G(X)$ of degree $d$ s.t. the locus $\mathcal{D}:Y=G(X)$ passes
through each of the points $(x_i,y_i)$ and, further, that for all
$1\leq k\leq m_i-1$
\[\deriv[k]{X}{}\bigg[G(X)^2\bigg]_{X=x_i}=\sderiv[k]{F}(x_i)\]
\end{theorem}

\begin{proof}
With the notation $h_k$ from Theorem \ref{thm:hkdiff} we note that we
have the $d+1$ equations
\begin{align}
G(x_i) &= h_0 (y_i)\\
\sderiv[k]{G}(x_i) &= h_k\big(
	y_i,\sderiv[1]{F}(x_i),\sderiv[2]{F}(x_i),\cdots,\sderiv[k]{F}(x_i)
	\big)
\end{align}
Our system of defining equations is in fact a system of $(d+1)$
inhomogeneous linear equations in the $(d+1)$ coefficients $g_i$ in
$G(X)=g_d+g_{d-1}X+\cdots+g_0 X^d$.
Define $r(i)=(1+\sum_{j=0}^{i-1} m_j)$, then
\begin{equation*}\tag{S}\label{eqn:system}
\begin{bmatrix}
\exvdm(\{(x_0,m_0),\cdots,(x_n,m_n)\}) & \big\vert & H
\end{bmatrix}
\end{equation*}
where $H$ is the $(d+1)\times 1$ matrix with
row $k+r(i)$ $(0\leq k<m_i)$ is
\[\begin{bmatrix}h_k\big(
	y_i,\sderiv[1]{F}(x_i),\sderiv[2]{F}(x_i),\cdots,\sderiv[k]{F}(x_i)
	\big)\end{bmatrix}\text{.}\]

In order to show that we have a solution it is necessary
to show that the rowspace of our system (\ref{eqn:system}) is equal to
that of $\exvdm$. For this solution to be unique we need these both in
turn to equal
$d+1$. In this respect it is necessary and sufficient to show that the
rowspace of
$\exvdm$ is $d+1$, i.e. that the determinant $\det \exvdm$ is nonzero.

Theorem \ref{thm:exvdm} implies that $\det \exvdm=0$ iff
$x_i=x_j$ for distinct $i,j$, but this does not happen in our list of
points, so the determinant is nonzero as required.
\end{proof}

\begin{example}
Consider the genus 2 curve $\mathcal{C}:Y^2=F(X)$ where
$$F(X)=X(X-1)(X-2)(X-5)(X-6)$$
and the list of points
$(0,0),\,(10,-120),\,(3,6),\,(3,6)$.
\[
\exvdm(\{(0,1),(10,1),(3,2)\})=
\begin{bmatrix}
1 &  0 &   0 &    0\\
1 & 10 & 100 & 1000\\
1 &  3 &   9 &   27\\
0 &  1 &   6 &   27
\end{bmatrix}
\quad H=\begin{bmatrix}
0\\
-120\\
6\\
3
\end{bmatrix}
\]
We obtain the system
\[
\begin{bmatrix}
1 &  0 &   0 &    0 & \big\vert &    0\\
1 & 10 & 100 & 1000 & \big\vert & -120\\
1 &  3 &   9 &   27 & \big\vert &    6\\
0 &  1 &   6 &   27 & \big\vert &    3
\end{bmatrix}
\mapsto
\begin{bmatrix}
1 & 0 & 0 & 0 & \big\vert &  0\\
0 & 1 & 0 & 0 & \big\vert & -2\\
0 & 0 & 1 & 0 & \big\vert & \frac{7}{3}\\
0 & 0 & 0 & 1 & \big\vert & -\frac{1}{3}
\end{bmatrix}
\]
\begin{figure}
 \begin{center}
  \epsfig{file=hell-4.ps,width=4in}
  \caption{Addition from \ref{eq:g2addexample}}\label{fig:g2addexample}
 \end{center}
\end{figure}
So define $G(X)=(0)+(-2)X+(\frac{7}{3})X^2+(-\frac{1}{3})X^3$. $G$
satisfies:
\begin{enumerate}
\item $G(0)=0$, $G(3)=6$, $G(10)=-120$
\item $\deriv{X}{}\big[G(X)^2\big]_{X=3}=36=\sderiv[1]{F}(3)$
\item $G(X)^2-F(X)=\frac{1}{9}X(X-1)(X-3)^2(X-6)(X-10)$
\end{enumerate}
We can deduce, as shown in the figure \ref{fig:g2addexample}, that
\begin{equation}\label{eq:g2addexample}
\{(0,0),\,(10,-120)\}+\{(3,6),\,(3,6)\}=\{(1,0),\,(6,0)\}
\end{equation}
\end{example}

\section{Overview of Addition Algorithm}

The algorithm is basically an implementation of calculating $\exvdm$
and $H$ and then solving the system \ref{eqn:system}. We then need to
factor $G(X)^2-F(X)$, but our task is simplified by the
fact that we already know many of the factors. In the genus 2 case,
with 4 points (counting multiplicity), we have at most a quadratic to
solve.

Before we plug our points into the algorithm, we need to make some
simplifications for `special cases'.
\begin{itemize}
	\item $\{A,B\}+\{C,D\}=\{A,C\}+\{B,D\}=\{A,D\}+\{B,C\}$ etc.
	\item When $Y^2=F(X)$, and the degree of $F$ is 5
	\begin{enumerate}
		\item $\{\infty,\infty\}$ and $\{(x,y),(x,-y)\}$ are
		all identified as $\mathfrak{O}$ s.t.
		$\mathfrak{A}+\mathfrak{O}=\mathfrak{A}$.
	\end{enumerate}
	\item When $Y^2=F(X)$, and the degree of $F$ is 6
	\begin{enumerate}
		\item We now have points $\infty^+,\infty^-$, so that
		$\{\infty^+,\infty^-\}$ and $\{(x,y),(x,-y)\}$ are
		all identified as $\mathfrak{O}$.
		\item The occurrence of $\infty^+$ or $\infty^-$
		produces restraints on the possible $G(X)$ which cannot
		be represented in the system from the theorem except by
		birationally transforming the curve.
	\end{enumerate}
	\item We drop occurrences of $\mathfrak{O}$ from our
	list of points, hence we need either 3 or 4 distinct
	points satisfying the hypothesis of the theorem
	or else we can return a result simply by rearranging
	the points we started with.
	\end{itemize}

The above algorithm, using $\exvdm$ is sufficient for genus 2, but for
higher genus the degree of $G$ is such that we have too many roots; in
this situation, we must solve for $p$ and $q$. The system of equations
is given by:
\begin{equation}
\frac{d^n}{dx^n}\big(q(x)y\big)\bigg|_{P_i}
 =\frac{d^n}{dx^n}p(x)\bigg|_{P_i}
\end{equation}
and the coefficients of $p$ and $q$ are given by:
\begin{equation}
 p_k:
 \begin{cases}
  0 & k<n\\
  \frac{k!}{(k-n)!}x^{k-n} & n\leq k\leq \partial p
 \end{cases}
\end{equation}
\begin{equation}
 q_k: 
  \sum_{i=0}^{k}\binom{n}{i}\frac{k!}{(k-i)!}x^{k-i}
  \frac{d^{n-i}y}{dx^{n-i}}\qquad 0\leq k\leq \partial q
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \input progs.tex

\appendix
\chapter{Summary of Key Programs}
\label{ap:progs}

We shall summarise a few of the main files of Maple programs
used in our computations. Even though they are designed for general
hyperelliptic curves of genus~$g$ (with an emphasis on genus~3),
they have nevertheless been placed in a subdirectory
of E.V.\ Flynn's genus~2 ftp directory, namely:

{\tt ftp://ftp.liv.ac.uk/pub/genus2/stubbs}

The files are:

\noindent
{\btt checkquarts.txt}
\par This lists the 27~relations of Theorem~\ref{thm:kummdefg3},
and checks that they are identities after expanding and
replacing each $y_i^2$ by $F(x_i)$. Note that the expressions are
far too large to expand initially, and so the file slices
the expressions into pieces and simplifies a portion at a time. 

\medskip\noindent
{\btt fast\_mult.txt} 
\par This gives a fast way of computing the coefficient of~$y_1 y_2$ 
in a product of the form $c_i c_j$, where $c_0,\ldots ,c_3$
are as in~(\ref{eq:g3kum}). This is helpful for finding the
relations in Theorem~\ref{thm:kummdefg3}.

\vfil\eject
\noindent
{\btt getkumm.txt}
\par This runs through monomials of the form $a_i b_j b_k$,
finds all linear combinations (four of them) for which
the coefficient of $y_1 y_2$ has $\wt$ at most~2, and
hence finds and outputs $c_0,\ldots ,c_3$ of~(\ref{eq:g3kum}).

\medskip\noindent
{\btt gpolyutil.txt}
\par This inputs the files mvmoreutils.txt, mvutils.txt, symm.txt, wreduce.txt,
and hence loads the package I have written for general routines
for hyperelliptic curves (with an emphasis on genus~3). The routines
which become available after the package is loaded dividing between
these four files, and are as follows.

{\obeylines 
\begin{alltt}{\btt{}checkonesym, conv, divalg, g3onesymmp, g3weight,
get_heaviest_terms, get_null_space, get_orbit,
get_weight_greaterthan, gethigh, initf, lc, lm,
multideg, mvcoeff, mvexps, mvmax, mvmono, mvnorm,
mvorder, naive_reduce_g3, onesym, onesymm, onesymminit,
permute_vars, simplify_poly_table_proc, try_simple_reduce_g3}\end{alltt}
}

\medskip\noindent
{\btt mvmoreutils.txt}
\par These are multivariate polynomial algorithms, such
as get\_null\_space, calculate\_hgs\_part, which extract parts of
highest weight, and which compute the space on induced by
the requirement that linear combinations of
a collection of expressions should have highest weight portion
set to zero.

\medskip\noindent
{\btt mvutils.txt}
\par General purpose routines for extracting portions of information,
such as extracting specified monomials, and division algorithms.

\medskip\noindent
{\btt symm.txt}
\par Programs for expressing symmetric functions as polynomials
of elementary symmetric functions.

\medskip\noindent
{\btt symmvar.txt}
\par Same as symm.txt, but with $a_0,\ldots ,a_g$ being used as
the choice for the elementary symmetric functions, and with
the output being made homogeneous.

\medskip\noindent
{\btt wreduce.txt}
\par The program naive\_reduce\_g3 performs reduction modulo
a power of delta, and g3weight computes the weight~$\wt$
of Definition~\ref{def:wt}.

\medskip\noindent {\btt hell/}
\par This directory contains files containing algorithms used in
calculating the $c_i$, as discussed in section (\ref{sec:calcci}).
Whilst many of the algorithms are available in files not contained in
this directory, the nature of the unpredictable interactions between
slightly different versions of the algorithms I experienced in using
\maple{maple} (especially in algorithms designed to work for slightly
different versions of \maple{maple} itself) suggests it is wiser to
use files that are known to work together. I have verified that the
files given in this directory can be used together to produce the
$c_i$ for genus 3.

\medskip\noindent
{\btt hell/thesis.v5}
\par This is a convenience file to read in the other files in this
directory into \maple{maple}.

\medskip\noindent
{\btt hell/hell.v5}
\par This contains the major (`front end') algorithms used in
producing the $c_i$. Of particular interest in this file are the
functions {\tt get\_monos}, which, given the degree of $F$, and the
required $w$- and $y$-weights, returns a list of monomials in the
$a_i$, $b_i$, and $f_i$ which sit in that weight partition;
{\tt distillation}, which, given the degree of $F$ and a list
of monomials, returns an array of polynomials sorted into levels of
the order of the pole on $\Theta$, after a distillation has been
performed; {\tt get\_nTheta} performs a triple-pass distillation given
the genus of the curve, the $w$- and $y$-weight partition to search
for polynomials, and the order of the pole on $\Theta$ that we want
our return polynomials to have.

\medskip\noindent
{\btt hell/mvutil.v5}
\par A hybrid of the functions available in files in the parent
directory.

\medskip\noindent
{\btt hell/pairsym.v5}
\par Some initial work in symmetric polynomials in pairs of variables.

\medskip\noindent
{\btt hell/sym.v5}
\par More work in symmetric polynomials.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\end{thebibliography}

\end{document}
