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\title
[Projective geometries arising from Elekes-Szab\'o problems]
{Projective geometries arising from Elekes-Szab\'o problems}
\author{Martin Bays and Emmanuel Breuillard}
\date{\today}
\begin{abstract}We generalise the Elekes-Szab\'o theorem to arbitrary arity
and dimension and characterise the complex algebraic varieties without power
saving. The characterisation involves certain algebraic subgroups of
commutative algebraic groups endowed with an extra structure arising from a
skew field of endomorphisms. We also extend the Erd\H{o}s-Szemer\'edi
sum-product phenomenon to elliptic curves. Our approach is based on
Hrushovski's framework of pseudo-finite dimensions and the abelian group
configuration theorem.
\end{abstract}
\maketitle
\section{Introduction}
%\subsection{Bounds on finite algebraic configurations}
Let $V \subset \C^n$ be an irreducible algebraic set over $\C$,
let $N \in \N$,
and let $X_i \subset \C$ with $|X_i| \leq N$, $i=1\ldots n$.
Then it is easy to see that
\[ |V \cap \prod_{i=1}^n X_i| \leq O_V(N^{\dim V}) .\]
Indeed, this follows inductively from the observation that there exists an
algebraic subset $W \subset V$ of lesser dimension and a co-ordinate
projection of the complement $V \setminus W \rightarrow \C^{\dim V}$ with
fibres of finite size bounded by a constant.
Say $V$ \defnstyle{admits no power-saving} if the exponent $\dim V$ is
optimal, i.e.\ if for no $\epsilon>0$ do we have a bound $|V \cap
\prod_{i=1}^n X_i| \leq O_{V,\epsilon}(N^{{\dim V}-\epsilon})$ as the $X_i$
vary among finite subsets of $\C$ of size $\leq N$.
% Note: configurations $X_i$ witnessing optimality are referred to as
% ``asymptotically near-optimal'' in ES-orchard.
In an influential paper Elekes and Szab\'o \cite{ES-groups} classified the
varieties which admit no power-saving in the case $n=3$. In order to state
their main theorem, we first need the following definition:
\begin{definition}
A \defnstyle{generically finite algebraic correspondence} between
irreducible algebraic varieties $V$ and $V'$ is a closed irreducible
subvariety of the product $\Gamma \subset V \times V'$ such that the
projections $\pi_V(\Gamma) \subset V$ and $\pi_{V'}(\Gamma) \subset V'$ are
Zariski dense, and $\dim(\Gamma) = \dim(V) = \dim(V')$.
\end{definition}
Suppose $W_1,\ldots ,W_n$ and $W_1',\ldots ,W_n'$ are irreducible algebraic
varieties, and $V \subset \prod_{i=1}^n W_i$ and $V' \subset \prod_{i=1}^n
W_i'$ are irreducible subvarieties.
Then we say $V$ and $V'$ are in \defnstyle{co-ordinatewise correspondence}
if there is a generically finite algebraic correspondence $\Gamma \subset V
\times V'$ and a permutation $\sigma \in \operatorname{Sym}(n)$ such that
for each $i$, the closure of the
projection $(\pi_i \times \pi'_{\sigma i})(\Gamma) \subset W_i \times
W_{\sigma i}'$ is a generically finite algebraic correspondence (between the
closure of $\pi_i(V)$ and the closure of $\pi'_{\sigma i}(V')$).
\begin{theorem}[Elekes-Szab\'o \cite{ES-groups}] \label{thm:ES}
An irreducible surface $V \subset \C^3$ admits no power-saving if and only
if either
\begin{enumerate}[(i)]\item $V \subset \C^3$ is in co-ordinatewise correspondence
with the graph $\Gamma = \{ (g,h,g+h) : g,h \in G \} \subset G^3$ of the
group operation of a 1-dimensional connected complex algebraic group
$G$,
\item \label{ES1triv} or $V$ projects to a curve, i.e.\
$\dim(\pi_{ij}(V))=1$ for some $i\neq j \in \{1,2,3\}$.
\end{enumerate}
\end{theorem}
%Fact[Hong Wang, Raz-Sharir-deZeeuw]:
% When 2 isn't optimal, 11/6 works.
% .
%(Could be that $1+\epsilon$ works for any $\epsilon>0$\ldots )
Here we generalise these results to arbitrary $n$ and $V \subset \C^n$.
\begin{definition}\label{special-one-dim}
An irreducible algebraic set $V \subset \C^n$ is \defnstyle{special} if it
is in co-ordinatewise correspondence with a product $\prod_i H_i \leq
\prod_i G_i^{n_i}$ of connected subgroups $H_i$ of powers $G_i^{n_i}$ of
1-dimensional complex algebraic groups, where $\sum_i n_i = n$.
\end{definition}
We prove:
\begin{theorem} \label{thm:main1}
An irreducible algebraic set $V \subset \C^n$ admits no power-saving if and
only if it is special.
\end{theorem}
The case of Theorem~\ref{thm:main1} with $V \subset \C^3$ and $\dim(V)=2$ is
precisely Theorem~\ref{thm:ES}. Indeed it is easy to verify that $V$ is special if and only if it is either of the form $(i)$ or of the form $(ii)$. The latter occurs exactly when the special subgroup $H \leq G^3$ can be taken to be a diagonal subgroup $\{x_i = x_j\}$, while the curve $\pi_{ij}(V)$ gives the correspondence.
The case $V \subset \C^4$ with $\dim(V)=3$ is a consequence of the results of
\cite{RSZ-ES4}.
A slightly stonger version of the case $V \subset \C^n$ with $\dim(V) = n-1$,
asking also for some uniformity in the power-saving (c.f.\
Remark~\ref{rem:gaps}), was conjectured by de Zeeuw in \cite[Conjecture~4.3]{deZeeuw-surveyER}.
The case $V \subset \C^4$ with $\dim(V) = 2$ solves \cite[Problem~4.4]{deZeeuw-surveyER}.
\begin{example}
$V := \{ (x,y,z,w) \in \C^4 : xzw=1=yz^2w^2 \}$ is special because it is a
subgroup of $(\C^*)^4$, and geometric progressions witness that it admits no
power-saving: setting $X = \{ 2^k : -M \leq k \leq M \}$, we find $|V \cap
X^4| \geq \Omega(M^2) \geq \Omega(|X|^2)$.
\end{example}
\begin{example}
Let $E \subset \P^2(\C)$ be an elliptic curve, say defined by $\{ y^2 =
x(x-1)(x-\lambda) \}$. Then taking $x$ co-ordinates yields a surface $V
\subset \C^3$ in co-ordinatewise correspondence with the graph $\Gamma_+
\subset E^3$ of the elliptic curve group law, and arithmetic progressions in
$E$ witness that $V$ admits no power-saving. This demonstrates the necessity
of taking correspondences in the definition of special.
To demonstrate the necessity of taking products, suppose
$E' \subset \P^2(\C)$ is another elliptic curve. Then taking $x$
co-ordinates
yields a 4-dimensional subvariety $W \subset \C^6$ in co-ordinatewise
correspondence with the product $\Gamma_+ \times \Gamma_{+'} \subset
E^3\times E'^3$ of the graphs of the two group laws, and again arithmetic
progressions witness that $W$ admits no power-saving. But if $E'$ is not
isogenous to $E$, then $W$ is not in co-ordinatewise correspondence with a
subgroup of a power of a single elliptic curve (see
Fact~\ref{fact:corrIsog}).
\end{example}
In fact we obtain a more general result, with arbitrary varieties in place of
the complex co-ordinates. Again, this generalises the corresponding result of
\cite{ES-groups}, who considered the case of a subvariety $V$ of $\C^d \times
\C^d \times \C^d$ of dimension $2d$ and with dominant projections to pairs of
co-ordinates, and showed that $V$ must be in correspondence with the graph of
multiplication of some algebraic group $G$. In \cite{breuillard-wang} it was
noted that this group must be commutative. Theorem~\ref{thm:main} below gives
a complete classification of the subvarieties without power saving, showing in
particular that the groups involved must be commutative. To state the result,
we first introduce the following definition.
\begin{definition} \label{defn:taucgp}
Let $W$ be a complex variety.
Let $C,\tau \in \N$ with $C\ge \tau$.
A finite subset $X \subset W$ is in \defnstyle{coarse $(C,\tau)$-general
position} in $W$ if for any proper irreducible complex closed subvariety $W'
\subsetneq W$ of complexity at most $C$,
we have $|W' \cap X| \leq |X|^{\frac 1\tau}$. When $C=\tau$ we will simply
say that $X$ is \defnstyle{$\tau$-cgp} in $W$.
\end{definition}
The notion of the complexity of a subvariety of a fixed variety is defined in
full generality in \ref{absVars} below. In the case that $W$ is affine, $W'
\subset W$ has complexity at most $C$ if it can be defined as the zero set of
polynomials of degree at most $C$.
Let $W_i$, $i=1,\ldots ,n$, be irreducible complex varieties each of dimension
$d$, and let $V \subset \prod_{i=1}^n W_i$ be an irreducible subvariety.
Now let $C,\tau \in \N$ and consider finite subsets $X_i \subset W_i$ with
$|X_i|\leq N^d$, $N \in \N$, and with each $X_i$ in coarse $(C,\tau)$-general
position in $W_i$.
As a straightforward consequence of coarse general position, if $\tau>d$ and
$C$ is sufficiently large depending on $V$ only, we will see in
Lemma~\ref{lem:power-saving} that we have a trivial bound $$|V \cap
\prod_{i=1}^n X_i| \leq O_V(N^{\dim(V)}).$$
We say that $V \subset \prod_i W_i$ \defnstyle{admits a power-saving by
$\epsilon>0$} if for some $C,\tau \in \N$ depending on $V$ only, this bound
can be improved to
$|V \cap \prod_{i=1}^n X_i| \leq O_{V,\epsilon}(N^{\dim(V)- \epsilon})$.
We say $V$ \defnstyle{admits no power-saving} if it does not admit a
power-saving by $\epsilon$ for any $\epsilon>0$.
It is easy to see that if $V$ admits no power-saving, then $\dim(V)$ must be
an integral multiple of $d$ (see Lemma~\ref{lem:power-saving}). In Theorem
\ref{thm:main} below we give a complete classification of the varieties with
no power-saving. To this end we introduce as earlier a notion of special
varieties, which generalises the previous definition and is slightly more
involved.
Let $G$ be a connected commutative complex algebraic group, and let $\End(G)$
be the ring of algebraic endomorphisms of $G$. We will denote by $\End^0(G)$
the $\Q$-algebra $\End^0(G):= \Q \otimes_{\Z} \End(G)$. For example, if $G$
is a torus $G=\mathbb{G}_m^r$, then $\End(G)=\Mat_r(\Z)$ and
$\End_0(G)=\Mat_r(\Q)$, and if $G=\mathbb{G}_a^r$ is a vector group, then
$\End(G)=\End^0(G)=\Mat_r(\C)$. In any case $\End(G)$ is a subring of
$\End^0(G)$.
\begin{definition} An algebraic subgroup of $G^n$ is called a
\defnstyle{special subgroup} if it has an ``$F$-structure'' for some
division subring $F$ of $\End^0(G)$, by which we mean that it is the
connected component of the kernel $\ker A \leq G^n$ of a matrix $A \in
\Mat_n(F \cap \End(G))$.
\end{definition}
For example $F$ could be trivial and equal to $\Q$, in which case the
corresponding special subgroups will be the connected components of subgroups
defined by arbitrary linear equations with integer coefficients in the $n$
co-ordinates of $G^n$.
\begin{remark} \label{remk:lieSpecial}
It will be convenient for us to express this condition in terms of the Lie
algebra $\Lie(H)$ of the subgroup $H \leq G^n$, which is defined as the
tangent space at the identity as a $\C$-vector space.
An algebraic endomorphism $\eta \in \End(G)$ induces by differentiation a
linear map $d\eta : \Lie(G) \rightarrow \Lie(G)$, making $\Lie(G)$ into an
$\End^0(G)$-module.
Then a subgroup $H \leq G^n$ is a special subgroup if and only if $\Lie(H) =
\Lie(G) \otimes_F J \leq \Lie(G)^n$
for some division subring $F \subset \End^0(G)$
and some $F$-subspace $J \leq F^n$
(where we make the obvious identifications between $\Lie(G)^n$, $\Lie(G^n)$
and $\Lie(G)
\otimes_F F^n$).
\end{remark}
\begin{definition}
An irreducible closed subvariety $V \subset \prod_{i=1}^n W_i$ of a product
of irreducible varieties is \defnstyle{special}
if it is in co-ordinatewise correspondence with a product $\prod_i H_i \leq
\prod_i G_i^{n_i}$ of special subgroups $H_i$ of powers $G_i^{n_i}$ of
commutative complex algebraic groups, where $\sum_i n_i = n$.
\end{definition}
This is consistent with Definition \ref{special-one-dim}, because when $G$ is one-dimensional, every connected algebraic subgroup of $G^n$ has an $F$-structure, where $F=End^0(G)$. See Lemma~\ref{lem:1dimSpecial} below.
We are now in a position to state our main theorem:
\begin{theorem} \label{thm:main}
Suppose $W_1,...,W_n$ are irreducible complex algebraic varieties of the
same dimension.
Then an irreducible subvariety $V \subset \prod_{i=1}^n W_i$ admits no
power-saving if and only if it is special.
\end{theorem}
\begin{example}
\newcommand{\HQ}{\mathcal{H}_\Q}
\newcommand{\HZ}{\mathcal{H}_\Z}
Let $G := (\C^\times)^4$. Then $\End^0(G) = \Q \otimes_{\Z} \End(G) \cong \Q
\otimes_{\Z} \Mat_4(\Z) \cong \Mat_4(\Q)$, the ring of $4\times 4$ rational
matrices. This is certainly not a division ring, but for example the
quaternion algebra
\[ \HQ = ( \Q[i,j,k] : i^2=j^2=k^2=-1;\; ij=k;\; jk=i;\; ki=j )\]
embeds in $\Mat_4(\Q)$ via the left multiplication representation.
This defines in particular an action of $\HZ = \Z[i,j,k] \subset \HQ$ on $G$
by endomorphisms given by a ring homomorphism $\alpha: \HZ \to \End(G), x
\mapsto \alpha_x$ defined by:
\begin{align*}
%n\cdot(a,b,c,d) &= (a^n,b^n,c^n,d^n); \\
\alpha_i(a,b,c,d) &= (b^{-1},a,d^{-1},c); \\
\alpha_j(a,b,c,d) &= (c^{-1},d,a,b^{-1}); \\
\alpha_k(a,b,c,d) &= (d^{-1},c^{-1},b,a).
\end{align*}
Then for instance $V := \{ (x,y,z_1,z_2,z_3) \in G^5 : z_1=x\cdot y,\; z_2=x
\cdot \alpha_i(y),\; z_3=x\cdot \alpha_j(y) \}$ is a special subgroup of
$G^5$.
To see that $V$ admits no power-saving, we can consider ``approximate
$\HZ$-submodules'': let $H_N := \{ n + mi + pj + qk : n,m,p,q \in
\{-N,\ldots ,N\} \}$, let $g \in G$ be generic (i.e.\ $\Q(g)$ has
transcendence degree $4$), and let $X_N := \alpha_{H_N}(g) = \{ \alpha_h(g)
: h \in H_N \} \subset G$. Since $\alpha_{\HZ}(g)$ is a finitely generated
subgroup of $G$, one can show (it is a consequence of Laurent's Mordell-Lang
theorem for tori, see Remark~\ref{remk:converseGp}) that for $W \subsetneq
G$ a proper Zariski-closed subvariety, $|W \cap \alpha_{\HZ}(g)|$ is finite
and bounded by a function of the complexity of $W$. Hence for all $\tau$,
for all sufficiently large $N$, we have that $X_N$ is $\tau$-$cgp$ in $G$.
But $\alpha_i(X_N) = X_N = \alpha_j(X_N)$, and so $|X_N^5 \cap V| \geq
\Omega(|X_N|^2)$.
We show in Subsection~\ref{subsect:sharp} that any special subgroup admits
no power-saving.
The argument of this example goes through for many groups $G$ (see
Remark~\ref{remk:converseGp}),
but extra complications arise with other groups - in particular in the case
of a power of the additive group $G=\mathbb{G}_a(\C)^d=(\C^d,+)$, where many
more division rings can arise and no Mordell-Lang type result holds.
\end{example}
\begin{remark} \label{rmk:mainCodim1}
In the situation of \cite[Theorem~27]{ES-groups} that $V \subset \C^d \times
\C^d \times \C^d $ projects dominantly with generically finite fibres to
each pair of co-ordinates, if $V$ is special then it is in co-ordinatewise
correspondence with the special subgroup $H_0 := \{x_1+x_2+x_3=0\}$ of a
$d$-dimensional commutative algebraic group $G$.
Indeed, we first obtain from Theorem~\ref{thm:main} that it is in
co-ordinatewise correspondence with the connected component of $\{\alpha_1
y_1+\alpha_2 y_2+\alpha_3 y_3=0\}$ for some self-isogenies $\alpha_i \in
\End(G)$; then, setting $x_i := \alpha_i y_i$, we see that this is in
co-ordinatewise correspondence with $H_0$.
Similarly for general $n$, if $\dim(V) = (n-1)d$ (as in \cite{RSZ-ES4} for
instance) then $\{\sum_ix_i=0\}$ is the only kind of special subgroup which
needs to be considered.
But if $V$ has higher codimension, endomorphisms are indispensable.
\end{remark}
\begin{remark}[Explicit power-saving] \label{rem:gaps}
We can consider strengthening Theorem~\ref{thm:main} by replacing the
condition that $V$ admits no power-saving with the condition that $V$ does
not admit a power-saving by $\eta$, where the ``gap'' $\eta$ is a constant
$\eta=\eta(d,n) > 0$.
The existence of such a gap in the case $n=3$ is part of
\cite[Main Theorem]{ES-groups}, and for $n=3$ and $d=1$ an explicit value of
$\eta=\frac16$ for this gap was found independently by Wang
\cite{Wang-gap} and Raz, Sharir, and de Zeeuw \cite{RSZ-ESR};
furthermore, \cite{RSZ-ES4} finds a gap of $\eta=\frac13$ for the case of
$n=4$ and $d=1$ (under a non-degeneracy assumption). For $n=3$ and $d$
arbitrary some explicit gaps were obtained by Wang and the second author
(see \cite{breuillard-wang}). None of the gaps are known to be optimal.
Our techniques for the general situation go via the abstraction of
combinatorial geometries and are not adapted to even proving the existence
of a gap, still less calculating one. However, in Section \ref{sec:warmup}
we work out the case of $n=3$, which does not require the full power of this
abstraction, and we obtain there \emph{an explicit gap $\eta=\frac{1}{16}$
for all $d$ (see Theorem \ref{cor:ES} below)} and also recover the
above-mentioned $\frac{1}{6}$ gap when $n=3$ and $d=1$.
\end{remark}
We draw as a corollary of Theorem~\ref{thm:main1} the following generalised
sum-product phenomenon.
\begin{corollary}[Generalised sum-product phenomenon]\label{gen-sum-product}
Let $(G_1,+_1)$ and $(G_2,+_2)$ be one-dimensional non-isogenous connected
complex algebraic groups,
and for $i=1,2$ let $f_i : G_i(\C) \rightarrow \C$ be a rational map.
Then there are $\epsilon,c>0$ such that if $A \subset \C$ is a finite set
lying in the range of each $f_i$, then setting $A_i=f_i^{-1}(A) \subset
G_i(\C)$ we have
\[ \max( |A_1 +_1 A_1|, |A_2 +_2 A_2| ) \geq c|A|^{1+\epsilon} .\]
% Note: can't get rid of c without requiring A_i to be sufficiently large,
% because it's possible for both A_i to be small finite subgroups.
\end{corollary}
\begin{remark} The usual sum-product phenomenon is the case $(G_1,+_1) =
(\C,+)$ and $(G_2,+_2) = (\C \setminus \{0\}, \cdot)$,
with $f_1$ and $f_2$ being the identity maps.
If instead $G_2 = E \subset \P^2(\C)$ is an elliptic curve defined by $\{
y^2 = x(x-1)(x-\lambda) \}$, then we may take $f_2$ to be the rational map
$[x:y:1] \mapsto x$. This case of the additive group and an elliptic curve
was previously considered for finite fields in
\cite{shparlinski-sumProdElliptic}.
The constant $c$ (which must depend on the $f_i$'s) is necessary as both
$A_i$'s could be finite subgroups of bounded order. We believe however that
the power-saving $\epsilon>0$ above is uniform over all group laws (and also
independent of the $f_i$'s); proving this would require establishing an
explicit gap in Theorem \ref{thm:main1} for $d=1$ and $n=6$. We do not tackle
this issue here.
\end{remark}
We also obtain the following result on intersections of subvarieties with
powers of an approximate subgroup, or just of a set with small doubling.
\begin{theorem} \label{thm:coherentApproxSubgroup}
Let $G$ be a commutative complex algebraic group.
Suppose $V$ is a subvariety of $G^n$ which is not a coset of a subgroup.
Then there are $N,\tau, \eps, \eta>0$ depending only on $G$ and the
complexity of $V$ such that if $A \subset G$ is a finite subset such that
$A-A$ is $\tau$-$cgp$ and $|A+A| \leq |A|^{1+\eps}$ and $|A| \geq N$, then
$|A^n \cap V| < |A|^{\frac{\dim(V)}{\dim(G)} - \eta}$.
\end{theorem}
Note that Theorem \ref{thm:main} yields right away that if no such $\eta>0$
exists, then $V$ must be special. So the point here is to show that under the
small doubling assumption the special $V$'s are in fact cosets of algebraic
subgroups. The result is reminiscent of the Larsen-Pink type estimates for
approximate groups (see \cite{hrushovski-wagner} \cite[Prop. 5.5]{hrushovski},
\cite[Thm 4.1.]{bgt}), with a stronger conclusion (the power-saving $\eta>0$)
and stronger hypothesis (coarse general position). This conclusion is also
reminiscent of results in Diophantine geometry of Manin-Mumford or
Mordell-Lang type, although our methods are completely unrelated; see Example
\ref{MM} for further comments in this direction.
\subsection{Method of proof}
The proof of our main results, Theorems \ref{thm:main1} and \ref{thm:main},
relies on an initial ultraproduct construction starting from a sequence of
finite subsets witnessing the absence of power-saving. This yields
pseudo-finite cartesian products. The field-theoretic algebraic closure
relation then induces an abstract projective geometry at the level of the
ultraproduct and we show, as a consequence of known incidence bounds
generalising the Szemer\'edi-Trotter theorem, that this geometry is modular,
i.e. satisfies the Veblen axiom of abstract projective geometries and can
therefore be co-ordinatised. The division rings appearing in Theorem
\ref{thm:main} arise that way. In the one-dimensional ($d=1$) case, the
projective geometries which embed in the geometry of algebraic closure in an
algebraically closed field were characterised in \cite{EH-projACF}, and in an
appendix we use similar techniques (primarily the abelian group configuration
theorem) to characterise them in the higher dimensional case. The main
combinatorial results above then follow.
Much of the strategy is an implementation of ideas due to Hrushovski appearing
in \cite{Hr-psfDims}, where he introduced the formalism of coarse
pseudo-finite dimension and outlined a proof of the original Elekes-Szab\'o
theorem in those terms.
More generally, our results are a consequence of specialising ideas of model
theory to this combinatorial setting. We use the conventions and language of
model theory throughout.
Nonetheless, our treatment requires very little model-theoretic background and
everything we need is described and recalled in Section \ref{sec:setup}. It is
also mostly self-contained, except for the use of the group configuration
theorem, recalled in Section \ref{sec:warmup}, and the Szemer\'edi-Trotter
type incidence bounds recalled in \S \ref{subsec:incidence}.
% terms of subgeometries of field-theoretic algebraic closure arising
%in ultraproducts from witnesses to optimality of the exponent. Hence they
%extend to sums of trivial geometries and projective geometries over division
%rings.
\subsection{Related work}
We remark here on how this paper relates to other recent works on applications
of model theory to similar problems.
In an unreleased work in progress, Hrushovski, Bukh, and Tsimmerman consider
expansion phenomena in pseudo-finite subsets of pseudo-finite fields of size
comparable to that of the non-standard prime field. This context is quite
different from that we consider, in particular because of the failure of
Szemer\'edi-Trotter in this regime, but there may be some overlap in
techniques; in particular, their analysis also proceeds via modularity and the
abelian group configuration theorem.
Meanwhile, Chernikov and Starchenko \cite{CS-ES} recently proved a version of
Theorem~\ref{thm:ES} in strongly minimal structures which are reducts of
distal structures. This direction of generalisation is orthogonal to the one
we consider here, where we restrict to the case of $ACF_0$ (this restriction
is used in Lemma~\ref{lem:SzT} and in Proposition~\ref{prop:converse}).
\subsection{Organisation of the paper} In Section \ref{sec:setup} we set up
our notation for the rest of the paper and present Hrushovski's notion of
pseudo-finite dimension of internal sets and its basic properties. This
section is entirely self-contained. We also recall the
Szemer\'edi-Trotter-type bounds for arbitrary varieties and recast them in
this language. In Section \ref{sec:warmup} we reprove the original
Elekes-Szab\'o theorem using the group configuration theorem and the formalism
of pseudo-finite dimensions. In higher dimensions we also recover the
commutativity of the ambient group and obtain an explicit power saving of
$\frac{1}{16}$. This section is not used in the proof of our main theorem, but
can be read as an example of the method, worked out in a special case. In
Section \ref{gpNecessity} we give a counter-example to the original
Elekes-Szab\'o theorem when the assumption of general position is removed.
Section \ref{sec:proj} contains the proof of the key point: the modularity of
the projective geometry associated to a variety without power-saving. In
Sections \ref{sec:varieties} and \ref{sec:asymptotic} we complete the proof of
Theorems \ref{thm:main1} and \ref{thm:main} modulo the result proven in the
Appendix. In particular we prove the converse (the ``if'' direction of the
theorems), which requires some information regarding division subrings of
matrices. We also derive Corollary \ref{gen-sum-product}. In Section
\ref{sec:subgroups} we prove Theorem \ref{thm:coherentApproxSubgroup} and draw
some connections with Diophantine geometry. Finally the appendix is devoted to
the higher-dimensional version of \cite{EH-projACF}.
\subsection{Acknowledgements}
Thanks to Mohammad Bardestani, Elisabeth Bouscaren, Ben Green, Martin Hils,
Udi Hrushovski, Jonathan Kirby, Oriol Serra, Pierre Simon and Hong Wang for
helpful conversations.
We would also like to thank the Institut Henri Poincar\'e and the organisers
of the
trimester ``Model theory, combinatorics and valued fields'', where some of the
work was done. The second author acknowledges support from ERC grant GeTeMo
no. 617129.
\setcounter{tocdepth}{1}
\tableofcontents
\section{The non-standard setup}\label{sec:setup}
In this preliminary section, we set up our notation and introduce the key
concepts, which will be used in the proof of the main results. We assume some
familiarity with the notions of first order languages, formulas and
ultraproducts as expounded for example in the first two chapters of
\cite{Marker-MT}. No other more sophisticated model-theoretical concepts will
be assumed.
\subsection{Coarse pseudo-finite dimension}
\label{subsect:psfDim}
We begin with a self-contained presentation of Hrushovski's formalism of
coarse pseudo-finite dimensions from
\cite{Hr-psfDims}, slightly adapted to our purposes.
\begin{parag}{\bf{Ultraproducts and internal sets.}} We will fix a
non-principal ultrafilter $\U$ on the set of natural numbers. We say that a
property of natural numbers holds for \defnstyle{$\U$-almost every} $s$ if
the set of natural numbers $s$ for which the property holds is an element of
$\U$.
We form the ultraproduct $K = \prod_{s \rightarrow \U} K_s$ of countably many
algebraically closed fields $K_s$, $s\ge 0$, which by definition is the
cartesian product $\prod_{s \ge 0} K_s$ quotiented by the equivalence relation
$(x_s)_s \sim (y_s)_s$ if and only if $x_s=y_s$ for $\U$-almost every $s \ge
0$. The field $K$ is also algebraically closed.
We will assume throughout {\bf internal characteristic zero}; namely, we
assume $\operatorname{char}(K_s) = 0$ for all $s$. This is required for the
incidence bounds used in Lemma~\ref{lem:SzT} below.
(See \cite[Corollary~5.6]{Hr-psfDims} for discussion on how it ought to be
possible to weaken this assumption).
In fact for our purposes it makes no difference to simply make the following
\begin{assumption}
We assume that $K_s=\C$ for all $s$.
\end{assumption}
We denote by $\stR := \R^\U$ the corresponding ultrapower of $\R$, and call
its elements \defnstyle{non-standard reals}. The real field $\R$ embeds
diagonally in $\stR$ and its elements are called standard reals. The order on
$\R$ extends to an order on $\stR$ by saying that $x0$,
$|z-\xi_s|<\epsilon$ holds for $\U$-almost every $s$.
Let $n$ be a positive integer. We say that a subset $X \subset K^n$ is
\defnstyle{internal} if $X = \prod_{s \rightarrow \U} X^{K_s}$ for some
subsets $X^{K_s} \subset K_s^n$.
\end{parag}
\begin{parag}{\bf{Saturation and compactness.}} \label{sat} A standard
property of ultraproducts over a countable index set is their
$\aleph_1$-compactness. Namely countable families of internal sets have the
finite intersection property. This means that for each positive integer $n$,
if $X_0 \supset X_1 \supset \ldots $ is a countable chain of internal
subsets of $K^n$ such that $\bigcap_{i \ge 0} X_i = \emptyset $, then $X_i =
\emptyset $ for some $i \ge 0$. Equivalently if an internal set $X \subset
K^n$ lies in the union of countably many internal sets, then it already lies
in the union of finitely many of them.
\end{parag}
\begin{parag}{\bf{Coarse pseudo-finite dimension.}}\label{basic-prop}
Throughout we will fix once and for all some infinite non-standard real $\xi
\in \stR$ with $\xi > \R$, which we call the \defnstyle{scaling constant}.
This choice corresponds to a choice of calibration for the large finite sets
involved in our main results. Given an internal set $X = \prod_{s
\rightarrow \U} X^{K_s} \subset K^n$, we define the non-standard cardinality
of $X$ by $|X| :=
\prod_{s \rightarrow \U} |X^{K_s}| \in \stR \cup \{\infty\}$ and its
\defnstyle{coarse pseudo-finite
dimension} $\bdl(X)$ by
\[ \bdl(X) := \st\left({\frac{\log |X|}{\log \xi}}\right) \in \R_{\geq 0} \cup
\{-\infty,\infty\} \]
(for the empty set we adopt the convention $\bdl(\emptyset)=\log(0) =
-\infty$).
\begin{example} Let $X^{K_s}:=\{(p,q) \in \N^2 : p+q~~0$ and a
$\emptyset$-definable set $Y \subset K^{n} \times K^m$ there is a
$\emptyset$-definable set $W \subset K^m$ such that $$ \{\b \in K^m :
\bdl(Y_{\b}) \geq \alpha+\epsilon\} \subset W \subset \{\b \in K^m :
\bdl(Y_{\b}) \geq \alpha\},$$ where $Y_{\b}$ is the fiber $\{\x \in K^n;
(\x,\b) \in Y\}$.
\end{definition}
It is always possible to force the continuity of $\bdl$ by enlarging the
language $\L$ to a new language $\L'$, which is still countable and for which
$\bdl$ becomes continuous. Indeed for each $q \in \Q$ we may add a predicate
to simulate the quantifier $\exists_{\ge \xi^q}$ of having ``at least $\xi^q$
solutions''. Explicitly, if $\xi=\lim_{s \to \U} \xi_s$ is as in the
definition of $\bdl$, let $\L_0 := \L$ and define $\L_{i+1}$ by adding to
$\L_i$ a new predicate $\psi_{\phi(\x,\y),q}(\y)$ for each formula
$\phi(\x,\y) \in \L_i$ and each $q \in \Q$, interpreted in $K_s$ by
$$\psi_{\phi(\x,\y),q}(K_s) := \{ \y : |\phi(K_s,\y)| \geq \xi_s^q\},$$ where
we have written $\phi(K_s,\y)$ for $\{\x : \phi(\x,\y) \textnormal{ holds in }
K_s\}$. So in the ultraproduct $K$ we have $\psi_{\phi(\x,\y),q}(K) = \{\y :
|\phi(K,\y)| \geq \xi^q\}$. Then we set $\L' := \cup_{i<\infty} \L_i$.
It is then clear that $\bdl$ is continuous once we replace $\L$ with $\L'$.
Indeed if $\alpha \in \R$ and $\epsilon>0$ we may pick a rational $q \in
(\alpha,\alpha+\epsilon)$. Then if $\b \in \psi_{\phi(\x,\y),q}(K)$ then
$|\phi(K,\b)| \geq \xi^q$ so $\bdl(\phi(K,\b)) \geq q > \alpha$, while if
$\bdl(\phi(K,\b)) \geq \alpha+\epsilon$ then $\bdl(\phi(K,\b)) \geq q$ and $\b
\in \psi_{\phi(\x,\y),q}(K)$.
\begin{remark}\label{C-def-cont} Note that the continuity property
automatically extends to definable sets with parameters. Namely if $Y$ is
assumed $C$-definable for some $C \subset K$, and $\bdl$ is continuous, we
may find a $C$-definable $W$ as in Definition \ref{cont-def}. Indeed there
is a finite tuple $\c_0 \in K^\ell$ for some $\ell \ge1$ with co-ordinates
in $C$ such that $Y=Y^0_{\c_0}=\{(\x,\y) : (\x,\y,\c_0) \in Y^0\}$ for some
$\emptyset$-definable set $Y^0 \in K^{n+m+\ell}$, so by continuity there is
$W^0$ a $\emptyset$-definable subset of $K^m$ such that $$ \{(\b,\c) \in
K^{m+\ell} : \bdl(Y_{\b,\c}) \leq \alpha\} \subset W^0 \subset \{(\b,\c) \in
K^{m+\ell} : \bdl(Y_{\b,\c}) \leq \alpha+\epsilon\}.$$ But now
$W:=W^0_{\c_0}$ is the desired $C$-definable set.
\end{remark}
%\begin{definition}
% We say that $\bdl$ has the continuity property (or is
% \defnstyle{continuous}) if given a
% formula $\phi(\x,\y) \in \L$ and $\alpha \in \R$, for any $\epsilon \in
% \R_{>0}$ there is a $\emptyset$-definable set $Y_{\phi,\alpha,\epsilon}$
% such that $$\textnormal{if }\b \in Y_{\phi,\alpha,\epsilon}, \textnormal{
% then }
% \bdl(\phi(K,\b)) < \alpha+\epsilon$$and $$\textnormal{if } \bdl(\phi(K,\b))
% \leq \alpha \textnormal{, then } \b
% \in Y_{\phi,\alpha,\epsilon},$$ where $\phi(K,\b)$ is the fiber $\{\x :
% \phi(\x,\b) \textnormal{ holds}\}$ of $\phi(K,K)$ above $\b$.
%\end{definition}
% Equivalently, the map from the type space to the 2-point compactification
% $S(\Th(K)) \rightarrow \R \cup\{-\infty,\infty\}: \tp(\b) \mapsto
% \bdl(\phi(\x,\b))$ is
% well-defined and continuous.
%It is always possible to force the continuity of $\bdl$ by enlarging the
%language $\L$ to a new language $\L'$, which is still countable and for which
%$\bdl$ becomes continuous. Indeed for each $q \in \Q$ we may add a predicate
%to simulate the quantifier $\exists_{<\xi^q}$ of having ``fewer than $\xi^q$
%solutions''. Explicitly, if $\xi=\lim_{s \to \U} \xi_{0,s}$ is as in the
%defintion of $\bdl$, let $\L_0 := \L$ and define $\L_{i+1}$ by adding to
%$\L_i$ a new internal set $X(\phi,q)=\prod_{s \to \U} X(\phi,q)_s$ for each
%formula $\phi:=\phi(\x,\y) \in \L_i$ and each $q \in \Q$, where $$X(\phi,q)_s
%:= \{ \y , |\phi(K,\y)| < \xi_{0,s}^q\}.$$ So in the ultraproduct $K$ we
%have $X(\phi,q)=\{\y : |\phi(K,\y)| < \xi^q\}$, where we have written
%$\phi(K,\y)$ for $\{\x : \phi(\x,\y) \textnormal{ holds}\}$. Then we set $\L'
%:= \cup_{i<\infty} \L_i$.
%It is then clear that $\bdl$ is continuous once we replace $\L$ with $\L'$.
%Indeed if $\alpha \in \R$ and $\epsilon>0$ we may pick a rational $q \in
%(\alpha,\alpha+\epsilon)$. Then if $\b \in X(\phi,q)$ then $|\phi(K,\b)| <
%\xi^q$ so $\bdl(\phi(K,\b)) \le q<\alpha+\epsilon$, while if
%$\bdl(\phi(K,\b)) \leq \alpha$ then $\bdl(\phi(K,\b)) < q$ and $\b \in
%X(\phi,q)$.
Continuity yields the following crucial properties, which are characteristic
of a dimension function; in particular, they are shared by transcendence
degree.
\begin{fact} \label{fact:dlContProps}
Let $\a,\b \in K^{< \infty}$ and let $C \subset K$ be countable and
$\phi(\x,\y)$ a formula in the language $\L$. If $\bdl$ is continuous (for
$\L$) then it is
\begin{enumerate}[(i)]\item \defnstyle{invariant}: if $\tp(\a)=\tp(\b)$,
then $\bdl(\phi(K,\a)) =
\bdl(\phi(K,\b))$,
\item \defnstyle{additive}:
\[ \bdl(\a\b/C)=\bdl(\b/C) + \bdl(\a/\b C).\]
\end{enumerate}
\end{fact}
Here as above $\phi(K,\a)$ denotes the definable set $\{\x : \phi(\x,\a)
\textnormal{ holds}\}$. We have used the convention $\alpha+\infty =
\infty+\alpha = \infty$, and $\a\b$ is a shorthand for $(\a,\b)$, the
concatenation of the tuples $\a$ and $\b$. Also we wrote $\b C$ for the union
of $C$ and the co-ordinates of $\b$.
\begin{proof} When $\a$ and $\b$ have the same type they belong to the same
definable sets, so $(i)$ is immediate from the continuity of $\bdl$. The
proof of $(ii)$ is given in \cite[Lemma~2.10]{Hr-psfDims}. We give it again
here for the reader's convenience. The idea is the following: if $Y$ is a
$C$-definable set in $K^n \times K^m$ containing $(\a,\b)$ and such that all
fibers $Y_{\b'}=Y \cap \pi_2^{-1}(\b')$ above the points $\b' \in \pi_2(Y)$
(where $\pi_2$ is the co-ordinate projection to $K^m$) have the same size,
then clearly $\bdl(Y) = \bdl(\pi_2(Y)) + \bdl(Y_{\b'})$. Now the continuity
property of $\bdl$ ensures that we can find a $Y$ with $\bdl(Y)$ close to
$\bdl(\a\b/C)$ and with all fibers of almost the same size. This shows
additivity.
We now give more details: by definition of the coarse dimension as an
infimum (see $(\ref{defdel})$), given $\epsilon>0$ we may find $C$-definable
sets $Y,Y' \subset K^n \times K^m$ such that $\a\b \in Y,Y'$ and
$\bdl(\a\b/C) \leq \bdl(Y) \leq \bdl(\a\b/C) +\epsilon$, $\bdl(\a/\b C) \leq
\bdl(Y'_{\b}) \leq \bdl(\a/\b C)+\epsilon$ and a $C$-definable set $Z\subset
K^m$ with $\b \in Z$ and $\bdl(\b/C) \leq \bdl(Z) \leq \bdl(\b/C)
+\epsilon$. Replacing $Y,Y'$ by $Y\cap Y' \cap \pi_2^{-1}(Z)$, we may assume
that $Y=Y'$ and $Z=\pi_2(Y)$. Now by continuity of $\bdl$ there is a
$C$-definable set $W \ni \b$ such that $|\bdl(Y_{\b'}) -
\bdl(Y_{\b})|<\epsilon$ for all $\b' \in W$. We may then further replace $Y$
by $Y \cap \pi_2^{-1}(W)$ and get to a situation where $\bdl(\a\b/C) \leq
\bdl(Y) \leq \bdl(\a\b/C) +\epsilon$, $\bdl(\b/C) \leq \bdl(\pi_2(Y)) \leq
\bdl(\b/C) +\epsilon$ and all fibers $Y_{\b'}$ for $\b'\in \pi_2(Y)$ have
$\bdl(\a/\b C)-\epsilon \leq \bdl(Y_{\b'}) \leq \bdl(\a/\b C)+\epsilon$. We
thus conclude that $|\bdl(\a\b/C) - \bdl(\b/C) -\bdl(\a/\b C)| \leq
3\epsilon$ as desired.
\end{proof}
%To handle the case of infinite $\bdl$ in this proof, we define
% $\infty-\infty := 0$.
% Let $\epsilon > 0$. By definition of the coarse dimension as an infimum
% (see $(\ref{defdel})$) we may find $\phi_1(x,y),\phi_2(x,y) \in
% \tp(\a,\b/C)$ and $\psi(y) \in \tp(\b/C)$ such that
% $|\bdl(\phi_1(K,K))-\bdl(\a\b/C)|, |\bdl(\phi_2(K,\b)) - \bdl(\a/C\b)|$ and
% $|\bdl(\psi(K)) - \bdl(\b/C)| $ are all at most $\epsilon$. We may further
% replace $\phi_1,\phi_2$ by $\phi(x,y):=\phi_1(x,y) \wedge \phi_2(x,y)
% \wedge \psi(y)$ and $\psi$ by $\exists x. \phi(x,y)$, only to get formulae
% in $\tp(\a,\b/C)$ and $\tp(\b,C)$ respectively with smaller or equal
% $\bdl$.
%Now by continuity of $\bdl$, there is a formula $\theta(y) \in \tp(\b/C)$
%such that if $\theta(\b')$ holds, then
%$|\bdl(\phi(K,\b')) - \bdl(\phi(K,\b))| < \epsilon$; replacing $\phi$ with
% $\phi(x,y) \wedge \theta(y)$, we may assume $\theta(y) \Leftrightarrow
% \exists x.\;
% \phi(x,y)$.
% Now if we have an estimate $c \leq |\phi(K,b)| \leq d$ and $e\Dashv \exists
% x.\;
% \phi(x,K)|$ with $c,d,e \in \R^\U$, then $ce \leq |\phi(K,K)| \leq de$,
% since we
% have the corresponding estimates with the corresponding finite sets in
% $\U$-many $K_s$.
% So we obtain $|\bdl(\phi(x,y)) - (\bdl(\phi(x,b)) + \bdl(\exists x.\;
% \phi(x,y)))| < \epsilon$.
% Then $|\bdl(ab/C) - (\bdl(a/Cb) + \bdl(b/C))| < 4\epsilon$.
% Since $\epsilon>0$ was arbitrary, we conclude.
%Recall from \cite[5.2]{Hr-approxSubgrps} the following basic properties of a
%pseudo-finite dimension $\dl$, including
%$\fdl$ and
%$\bdl$:
%Lemma{lem:dlProps}:--
% **(i)
% $\dl$ is subadditive: if $f$ is definable and $\dl(f^{-1}(f(x))) \leq
% \xi$
% for all $x \in X$, then $\dl(X) \leq \dl(f(X)) + \xi$.
\begin{remark}
\newcommand{\Km}{\mathbb{K}}
We briefly remark in passing for the model-theoretically inclined reader that
a more sophisticated setup is also available, which in some ways is more
satisfactory than that described above.
Working directly in a countable ultrapower with only $\aleph_1$-compactness,
as we have in this section, has the consequence that we must pick a
countable language to work with. In our applications we will have no real
control over the definable sets and can expect no tameness, so having to
make this choice is something of a distraction. An alternative would be to
define $K$ as above but in a language $\L_\opint$ which includes all
internal sets as predicates, and then to take a $\kappa$-saturated
$\kappa$-strongly homogeneous elementary extension $\Km$, for a cardinal
$\kappa$ which is larger than any parameter set we wish to consider.
There is then a unique way to define $\bdl(\phi(\x,\a))$ for $\phi \in
\L_\opint$ and $\a \in \Km^{<\omega}$ such that $\bdl$ is continuous and
extends the original definition in the case $\a \in K^{<\omega}$.
Namely, $\bdl(\phi(\x,\a)) := \sup \{ q \in \Q : \Km \vDash \exists_{\geq
\xi^q} \x.\; \phi(\x,\a) \}$, where $\exists_{\geq \xi^q} \x.\; \phi(\x,\y)$
denotes an $\L_\opint$ formula with free variables $\y$ such that $K \vDash
\exists_{\geq \xi^q} \x.\; \phi(\x,\b)$ if and only if $| \phi(K,\b) | \geq
\xi^q$, for $\b \in K^{<\omega}$.
(This is parallel to the way one defines dimension on an elementary
extension of a Zariski structure.)
Here, continuity is meant in the sense of Definition~\ref{cont-def} - or
equivalently, that the map from the type space to the 2-point
compactification $S_{\y}(\emptyset) \rightarrow \R \cup\{-\infty,\infty\}:
\tp(\b) \mapsto \bdl(\phi(\x,\b))$ is well-defined and continuous.
We can then work with elements of $\Km$ in order to analyse the internal
subsets of $K$. We will not use this alternative presentation, but some
readers may prefer to pretend that we do so throughout.
\end{remark}
\end{parag}
\begin{parag}{\bf Algebraic independence and transcendence
degree.}\label{notn:baseField} At the heart of the combinatorial results of
this paper lies the interplay between combinatorics (via the coarse
pseudo-finite dimension $\bdl$) and algebraic geometry (via the notion of
algebraic dimension, or transcendence degree). To this effect we will fix a
\defnstyle{base field} $C_0$ and assume it is \emph{countable and
algebraically closed} and contained in $K$. We will then have to consider
the subclass of definable sets that are $C_0$-definable using only the
language of rings $\L_{ring}$. In the applications $C_0$ will be the
algebraic closure of the field of definition of the variety. As is
well-known, in an algebraically closed field, the sets that are
$C_0$-definable in $\L_{ring}$ coincide with the so-called constructible
sets of algebraic geometry defined over $C_0$, namely solutions of finitely
many polynomial equations and inequations with coefficients in $C_0$. After
enlarging $\L$ if necessary we can make the following
\begin{assumption}
We assume that $\L$ contains a constant symbol for each element of $C_0$.
\end{assumption}
\begin{notation*}[$0$ superscript] We will use a superscript $0$, e.g.\
$\tp^0$, to indicate that we work in the
structure $(K;+,\cdot,(c)_{c \in C_0})$ of $K$ as an algebraically closed
field extension of the base field $C_0$, rather than in the full language
$\L$. For example for $\a,\b \in K^{< \infty}$ and $C \subset K^{<\infty}$,
saying that
$\tp^0(\a/C)=\tp^0(\b/C)$ means that they satisfy the same
polynomial equations over the field $C_0(C)$ generated by $C_0$ and the
co-ordinates of all tuples belonging to $C$, i.e.\ for $f \in
C_0(C)[\tuple{X}]$, $f(\a)=0$
if and only if $f(\b)=0$.
\end{notation*}
\begin{notation*}[algebraic closure $\acl^0$] Similarly for a subset $A
\subset K^{< \infty}$ we denote by $\acl^0(A)$ the field-theoretic
algebraic closure in $K$ of the subfield $C_0(A)$ generated by $C_0$ and
the co-ordinates of the elements of $A$.
\end{notation*}
When there is no superscript, we work in the full language $\L$.
% As usual, $\dcl(A)$ is the union of all singleton subsets of $K$ definable
% over $A$.
\begin{notation*}[transcendence degree $\dimo$] We write $\dimo$ for the
dimension with respect to $\acl^0$, i.e.\ for $A,B \subset K^{< \infty}$
we set:
$$\dimo(A/B) := \trd(C_0(AB)/C_0(B)),$$ where $\trd$ denotes the transcendence
degree, and $C_0(B)$ the field extension of $C_0$ generated by $B$ and $AB$
is short for $A \cup B$.
\end{notation*}
Note that, just like $\bdl$, \emph{$\dimo$ is additive}: if $\a,\b \in
K^{<\infty}$ and $C \subset K$, then \begin{equation}\label{add-dim0}\dimo(\a
\b / C) = \dimo(\a / \b C) + \dimo(\b / C),
\end{equation}
where, as earlier, $\b C$ is short for the union of $C$ and the co-ordinates
of $\b$.
Note finally that clearly $\dimo(A/B) = \dimo(A/\acl^0(B))$.
\begin{notation*}[independence $\ind^0_C$]\label{ind} If $A,B,C$ are subsets
of tuples of $K$, we will say that $A$ is algebraically independent of $B$
over $C$ and write $A \ind^0_C B$ if $$\dimo(A/BC) = \dimo(A/C),$$ i.e.\
if $C_0(A)$ is algebraically independent from $C_0(B)$ over $C_0(C)$.
This is clearly a symmetric relation, namely $A \ind^0_C B$ if and only if $B
\ind^0_C A$.
\end{notation*}
% The base field $C_0$ will not be fixed throughout the text, but we will be
% explicit when we vary it.
% Given a countable set of parameters $A$, by \defnstyle{base-change to $A$}
% we refer to the operation of adding the elements of $C_0(A)$ as constants
% to
% the language, so then $C_0(A) \leq \dcl(\emptyset )$, and setting $C_0(A)$
% as the base
% field.
%\begin{notation} \label{notn:dind}
% $A \dind_C B$ means that for any tuple $\a \in A^{< \infty} =
% \bigcup_{i\in\N}A^i$, we have
% $\bdl(\a/BC) = \bdl(\a/C)$.
%\end{notation}
%This may fail to be symmetric due to $\bdl$ taking value $\infty$.
\begin{notation*} For $A \subset K^{< \infty}$, we write
\begin{equation}\label{closalg} \acl^0(A)^{< \infty} := \bigcup_{n\ge 1}
(\acl^0(A))^{n} \subset K^{< \infty} ,
\end{equation} for the set of tuples algebraic over $A$. Note that this is
also
the set of tuples with finite orbit under the group of field automorphisms
$\Aut(K/C_0(A))$ fixing $C_0(A)$ pointwise.
\end{notation*}
\end{parag}
% Occasionally we consider a finite tuple from $\Keqz$ as an element of
% $\Keqz$, in the obvious way.
\begin{parag}{\bf Coarse dimension of an algebraic tuple.} Let $C \subset K^{<
\infty}$ be a countable subset. If a tuple $\a$ belongs to $\acl^0(C)^{<
\infty}$, then it is contained in a finite $C$-definable set, namely the
Galois orbit of $\a$ over $C_0(C)$. In particular, since $\xi > \R$ we have
$\bdl(\a/C) = 0$. So
\[ \a \in \acl^0(C)^{< \infty} \Rightarrow \bdl(\a/C) = 0 .\]
We also record here the following generalisation of this observation, which
will be used in the proof of Proposition \ref{lem:cgpCoherentLinearity}. For
any $\a \in K^{< \infty}$ and countable $C \subset K$:
\begin{equation}\label{alg-ineq} \bdl(\a/C)=\bdl(\a/\acl^0(C)).
\end{equation}
Indeed, first we have $\bdl(\a/C)\geq \bdl(\a/\acl^0(C))$ by $(\ref{mono2})$.
For the opposite inequality it is enough to show that if $\b \in \acl^0(C)$,
then $\bdl(\a/C) \leq \bdl(\a/\b)$. To see this, note that $\bdl(\b/C)=0$ by
the above remark, and thus by additivity $(\ref{fact:dlContProps}.ii)$
$$\bdl(\a/C) \leq \bdl(\a\b/C) = \bdl(\a/\b C)+\bdl(\b/C)= \bdl(\a/\b C) \leq
\bdl(\a/\b).$$
\end{parag}
%\begin{remark} \label{notn:Keqz0}
% Let $\Keqz := K^{< \infty} = \bigcup_{i \in \N}K^i$ be the set of finite
% tuples from $K$. By elimination of imaginaries in $ACF$, any
% $ACF$-imaginary
% of $K$ is $ACF$-interdefinable with an element of $\Keqz$, so we could just
% as well define $\Keqz$ to be the union of the $ACF$-imaginary sorts, which
% explains the notation. In particular, any $K$-point of an algebraic variety
% over $K$ is $ACF$-interdefinable with an element of $\Keqz$.
%\end{remark}
%Lemma{lem:CS}:
% If $f : X \rightarrow Y $ is definable, then the self-fibre-product of $X$
% with itself with respect to $f$, $X \times_f X := \{ (x,y) | x \in X, y
% \in X, f(x)=f(y) \}$, satisfies $\bdl(X \times_f X) \geq 2\bdl(X) -
% \bdl(f(X))$.
% .
%Proof:
% When $X$ is definable, this follows from a Cauchy-Schwarz argument - see
% \cite[2.2]{Hr-psfDims}. In general,
% for any $\epsilon>0$ in $\R$, there exists $X' \supset X$ such
% that $\bdl(f(X')) < \bdl(f(X))+\epsilon$, and then for any definable $X''$
% with $X \subset X'' \subset X'$, $\bdl(X'' \times_f X'') \geq 2\bdl(X'') -
% \bdl(f(X'') >
% 2\bdl(X) - \bdl(f(X)) - \epsilon$.
% So $\bdl(X \times_f X) \geq 2\bdl(X) - \bdl(f(X)) - \epsilon$.
% .
%
%{\em what's the best version of this for fine dimension and
%$\Wedge$-definable $X$?}
\begin{parag}{\bf Locus of a tuple.} If $\a \in K^n$ and $C \subset K$, we
define the \defnstyle{locus} of $\a$ over $C_0(C)$, denoted by
$\loc^0(\a/C)$, to be the smallest Zariski-closed subset $V \subset K^n$
such that $\a \in V$ and $V$ is defined by the vanishing of polynomials with
coefficients in $C_0(C)$. We also write $\loc^0(\a)$ for
$\loc^0(\a/\emptyset)$.
Note that by definition $\loc^0(\a/C)$ is irreducible over $C_0(C)$, i.e.\ it
cannot be written as a finite union of more than one proper Zariski-closed
subset of $K^n$ defined over $C_0(C)$, but it may not be absolutely
irreducible (i.e.\ irreducible over $K$). However each absolutely irreducible
component is defined over some finite algebraic extension. In particular
$\loc^0(\a/\acl^0(C))$\emph{ is an absolutely irreducible component of}
$\loc^0(\a/C)$, and
\begin{equation}\label{dim-loc}\dimo(\a/C)=\dim(\loc^0(\a/C))=\dim(\loc^0(\a/\acl^0(C))).
\end{equation}
\end{parag}
\begin{parag}{\bf Abstract varieties.} \label{absVars}
Our setup is adapted to working with tuples of elements of $K$, but in our
applications we will want to work with points of algebraic groups and of
general abstract algebraic varieties. We explain here how we bridge this gap
using standard notions from the model theory of algebraically closed fields,
as described in \cite{Pillay-ACF} or \cite[7.4]{Marker-MT}.
We adopt the convention that varieties are always separated, but not
necessarily irreducible.
If $V$ is an algebraic variety over an algebraically closed subfield $C \leq
K$, then $V$ admits a cover by finitely many affine open subvarieties over
$C$; that is, there are open subvarieties $V_i \subset V$ and (closed)
affine subvarieties $U_i \subset \mathbb{A}^{n_i}$ and isomorphisms $f_i :
V_i \rightarrow U_i$ over $C$, such that $V = \bigcup_i V_i$.
Then $V(K)$ can be identified with the quotient of the disjoint union of the
$V_i(K)$ by the equivalence relation of representing the same point of
$V(K)$.
Now $ACF_0$, the theory of algebraically closed fields of characteristic
zero, admits elimination of imaginaries, which exactly means that such a
quotient is in definable bijection over $C$ with a definable (i.e.\
constructible) subset of $K^n$ for some $n$. We refer to
\cite[Remark~3.10(iii), Lemma~1.7]{Pillay-ACF} for details of this
construction.
In this way, we embed $V(K)$ as a subset of $K^n$.
Note that this embedding is not continuous.
The precise embedding depends on our choice of cover. However, if $W \subset
V$ is another subvariety and $f : W \rightarrow U \subset K^m$ is an
isomorphism over $C$ with an affine variety, and if $a \in W(K) \subset V(K)
\subset K^n$, then the subfield of $K$ generated over $C$ by the
co-ordinates of $a$ according to our embedding of $V(K)$ in $K^n$ and those
according to $f$ are equal, $C(a) = C(f(a))$. In particular, for $a \in
V(K)$ the subfield $C(a) \leq K$ does not depend on our choice of cover.
For $\tau \in \N$, we say that the \defnstyle{complexity} of a closed
subvariety $W \subset V$ is at most $\tau$ if for each $i$ the affine
variety $f_i(W\cap V_i) \subset U_i \subset \mathbb{A}^{n_i}$ can be defined
as the set of common zeros of a collection of polynomials each of degree at
most $\tau$.
Note that the family of subvarieties of $V$ of complexity at most $\tau$
forms a definable family; that is, there is $m \in \N$ and a constructible
set $X \subset V \times K^m$ over $C$ such that every subvariety of $V$ over
$K$ of complexity at most $\tau$ is of the form $X(\b) = \{ v : (v,\b) \in X
\}$ for some $\b \in K^m$. In fact this is the only property we require of
the notion of complexity.
% \mbcomment{Maybe we should call this ``relative complexity'', or
% ``complexity in $V$'', to avoid conflict with BGT?}
% {\bf EB: I wouldn't bother.}
\end{parag}
\begin{parag}{\bf Generic elements.} Let $V$ as before be an algebraic variety
over an algebraically closed $C \leq K$. For $a \in V(K)$ and $C' \leq K$ an
algebraically closed subfield containing $C$, we define the
\defnstyle{locus} of $a$ over $C'$ within $V$, $\locus^V(a/C')$, to be the
smallest Zariski-closed subvariety of $V$ defined over $C'$ and containing
$a$. If $V(K) \subset K^m$ is affine and defined over $C$, then
$\locus^V(a/C)=\loc^0(a/C)$.
If $V$ is irreducible, a point $a \in V(K)$ of $V$ is \defnstyle{generic} if
it is contained in no proper closed subvariety over $C$, i.e.\
$\locus^V(a/C) = V$; equivalently, $\trd(a/C) = \dim(V)$.
\begin{remark} \label{rmk:corrGeneric}
If $V \subset \prod_i W_i$ and $V' \subset \prod_i W_i'$ are closed
subvarieties where $V,V',$ $W_i,W_i'$ are irreducible varieties over $C_0$,
then $V$ and $V'$ are in co-ordinatewise correspondence if and only if they
have generics $\a \in V(K)$ and $\a' \in V'(K)$ such that $a_i \in W_i(K)$
and $a'_i \in W'_i(K)$ are generic and for some permutation $\sigma \in
\operatorname{Sym}(n)$, we have $\acl^0(a_i) = \acl^0(a_{\sigma i}')$.
Indeed, $\loc^{0}(a_i,a_{\sigma i}') \subset W_i\times W_{\sigma i}'$ is
then a generically finite algebraic correspondence between $\loc^0(a_i)$ and
$\loc^0(a_{\sigma i})$, as required.
\end{remark}
\end{parag}
\begin{parag}{\bf Canonical base.}\label{parag:canonical} In the proof of our
main theorems, it will be crucial to understand the dimensions of certain
families of varieties. The right concept for this (which serves a similar
purpose as the concept of Hilbert scheme in classical algebraic geometry) is
the notion of canonical base.
Recall that the \emph{field of definition} of a Zariski-closed subset $V
\subset K^n$ is
the smallest field $k$ over which $V$ is defined; equivalently (since
$\operatorname{char}(K)=0$), $k$ is such that a field automorphism $\sigma
\in \Aut(K)$
fixes $V$ setwise if and only if it fixes $k$ pointwise.
Given $\a \in K^{n}$ and $C \subset K^{<\infty}$ let $k\leq K$ be the field
of definition of the absolutely irreducible Zariski-closed subset
$\loc^0(\a/\acl^0(C))$ of $K^n$. A tuple $\d\in K^{<\infty}$ is said to be a
\defnstyle{canonical base} of $\a$ over $C$ if its co-ordinates together
with $C_0$ generate the subfield of $K$ generated by $C_0$ and $k$.
Clearly if $\d \in K^{<\infty}$ is a canonical base of $\a$ over $C$ then it
is a canonical base of $\a$ over $\acl^0(C)$ and conversely. Furthermore $\d
\in \acl^0(C)$ and since $\loc^0(\a/\acl^0(C))$ is defined over $C_0(\d)$ we
have:
$$\loc^0(\a/\acl^0(C)) = \loc^0(\a/\d)$$ so $\loc^0(\a/\d)$ is (absolutely)
irreducible and
$$\dimo(\a/\d) = \dimo(\a/C) = \dimo(\a/\acl^0(C)) = \dimo(\a/C \d),$$
in other words: $\a \ind^0_{\d} C$. In the proof of Proposition
\ref{lem:cgpCoherentLinearity} below we shall require the following fact.
%\begin{lemma} \label{lem:cb}
% Let $\a \in K^{<\infty}$ and $C \subset K^{< \infty}$. Let $\d \in
% K^{<\infty}$ be a canonical base of $\a$ over $C$. Let $\d' \in
% K^{<\infty}$. Assume that $\tp^0(\d/\a)= \tp^0(\d'/\a)$ and that $\a
% \ind^0_{\d} \d'$. Then $\d=\d'$.
%\end{lemma}
%\begin{proof} Let $\sigma \in \Aut(K/C_0(\a))$ with $\sigma(\d)=\d'$.
%Since $\a \ind^0_{\d} \d'$ the varieties $\loc^0(\a/\d \d') \subset
%\loc^0(\a/\d)$ have the same dimension, hence coincide because
%$\loc^0(\a/\d)$ is irreducible given that $\d$ is a canonical base for $\a$.
%But $\loc^0(\a/\d \d') \subset \loc^0(\a/\d') = \sigma(\loc^0(\a/\d))$,
% and so since they have the same dimension in fact
% $\loc^0(\a/\d) = \sigma(\loc^0(\a/\d))$. So $\sigma$ fixes the field of
% definition of $\loc^0(\a/\d)$ pointwise,
% and so in particular $\sigma(\d)=\d$ and $\d' = \d$.
%\end{proof}
\begin{lemma} \label{lem:cb}
Let $\a \in K^{<\infty}$ and $C \subset K^{< \infty}$. Let $\d \in
K^{<\infty}$ be a canonical base of $\a$ over $C$ and $V:=\loc^0(\a\d)$. Let
$\d_1,\d_2 \in \tp^0(\d)(K)$ and $\a' \in K^{< \infty}$ such that $\a'\d_i
\in V$. Then either $\dimo(\a'/\d_1\d_2) < \dimo(\a/\d)$, or $\d_1=\d_2$.
\end{lemma}
\begin{proof} Note that $$\dimo(\a'/\d_1\d_2) \le
\dimo(\a'/\d_1)=\dimo(\a'\d_1)-\dimo(\d_1)\le \dim V - \dimo(\d) =
\dimo(\a/\d).$$ So if $\dimo(\a'/\d_1\d_2) \ge \dimo(\a/\d)$, then the above
inequalities are equalities and in particular $\dimo(\a'\d_i)=\dimo(\a\d)$
for each $i$. Since $V$ is irreducible we obtain $V=\loc^0(\a'\d_i)$. Hence
there exist $\sigma_i \in \Aut(K/C_0)$ with $\sigma_i(\a)=\a'$ and
$\sigma_i(\d)=\d_i$.
Since $\d$ is a canonical base for $\a$ over $C$, $C_0(\d)$ is the field of
definition of $\loc^0(\a/\d)$. Hence $C_0(\d_i)$ is the field of definition
of $\loc^0(\a'/\d_i)$, and thus $\d_i$ is a canonical base of $\a'$ over
$\d_i$. In particular $\loc^0(\a'/\d_i)=\sigma_i(\loc^0(\a/\d))$ is
irreducible. Since $\loc^0(\a'/\d_1\d_2) \subset \loc^0(\a'/\d_i)$ have the
same dimension, we conclude that $\loc^0(\a'/\d_1\d_2)=\loc^0(\a'/\d_i)$.
In particular $\loc^0(\a'/\d_1)= \loc^0(\a'/\d_2)$. Setting $\sigma=\sigma_2
\sigma_1^{-1}$, we get $\sigma(\a')=\a'$, $\sigma(\d_1)=\d_2$ and
$\sigma(\loc^0(\a'/\d_1))=\loc^0(\a'/\d_2)=\loc^0(\a'/\d_1)$. Hence $\sigma$
fixes $\loc^0(\a'/\d_1)$ setwise. It must fix its field of definition
$C_0(\d_1)$ pointwise. Hence $\sigma(\d_1)=\d_1$, and $\d_2=\d_1$ as claimed.
\end{proof}
\end{parag}
%\begin{definition}
% If $a \in \Keqz$ and $C \subset \Keqz$,
% let $\Cb^0(a/C) \in \Keqz$ be the canonical base of the strong type
% $\stp^0(a/C)$ of $a$ over $C$ in the reduct to $ACF$ with parameters for
% $C_0$.
% So $\Cb^0(a/C)$ is well-defined up to $\dcl^0$-interdefinability.
%\end{definition}
%\begin{remark}
% Recall that the field of definition of a Zariski-closed subset $V \subset
% K^n$ is
% the smallest field $k$ over which $V$ is defined; equivalently (since
% $\operatorname{char}(K)=0$), $k$ is such that a field automorphism $\sigma
% \in \Aut(K)$
% fixes $V$ setwise if and only if it fixes $k$ pointwise.
% If $a \in \Keqz$ and $C \subset \Keqz$, then $\Cb^0(a/C)$ is any tuple of
% generators over $C_0$ for the field of definition of $\loc^0(a/\acl^0(C))$.
%\end{remark}
%The following is all we require of canonical bases. We give proofs directly
%from the algebraic characterisation of the previous remark.
%\begin{lemma} \label{lem:cb}
% Let $a,d \in \Keqz$ and $C \subset \Keqz$.
% \begin{enumerate}[(i)]\item $A := \acl^0(\Cb^0(a/C))$ is the smallest
% $\acl^0$-closed subset of
% $\acl^0(C)$ such that $a \ind^0_A C$.
% \item Suppose $d = \Cb^0(a/d)$ and $d' \equiv ^0_{a} d$ and $d' \neq d$.
% Then $a \nind^0_d d'$.
% \end{enumerate}
%\end{lemma}
%\begin{proof}
% \begin{enumerate}[(i)]\item Model theoretically, this is standard.
%
% Algebraically, for any $\acl^0$-closed $A' \subset C$,
% we have: $a \ind^0_{A'} C$ if and only if $V := \loc^0(a/\acl^0(C)) =
% \loc^0(a/A')$
% if and only if $V$ is defined over $A'$
% if and only if $C_0(\Cb^0(a/C)) \subset A'$
% if and only if $A \subset A'$.
% \item
% Model theoretically, this is because $\tp(a/d)$ is not parallel to its
% conjugate $\tp(a/d')$.
%
% Algebraically, let $\sigma \in \Aut(K/C_0(a))$ with $\sigma(d)=d'$.
% Then if $a \ind^0_d d'$,
% then $\loc^0(a/d) = \loc^0(a/dd') \subset \loc^0(a/d') =
% \sigma(\loc^0(a/d))$,
% and so since they have the same dimension in fact
% $\loc^0(a/d) = \sigma(\loc^0(a/d))$,
% so $\sigma$ fixes the field of definition of $\loc^0(a/d)$ pointwise,
% and so in particular $\sigma(d)=d$, contradicting $d' \neq d$.
% \end{enumerate}
%\end{proof}
%
\begin{parag}{\bf Isogenies.}
We say that commutative algebraic groups $G_1,G_2$ are \defnstyle{isogenous}
if there exists an \defnstyle{isogeny} $\theta : G_1 \twoheadrightarrow G_2$;
that is, a surjective algebraic group homomorphism with finite kernel. The
relation of being isogenous is an equivalence relation.
We will apply in multiple places the following useful criterion for the
existence of an isogeny.
\begin{fact} \label{fact:corrIsog}
Let $(G;\times)$ and $(G';+)$ be connected algebraic groups over an
algebraically closed field $C_0$ of characteristic zero.
Suppose the graphs $\Gamma_{\times}$ and $\Gamma_+$ of the group operations
are in co-ordinatewise correspondence, and $G'$ is commutative. Then $G$ is
also commutative, and is isogenous to $G'$.
Moreover, if $(g,h) \in G^2(K)$ and $(g',h') \in (G')^2(K)$ are each
generic,
and if $\acl^0(g) = \acl^0(g')$ and $\acl^0(h) = \acl^0(h')$ and
$\acl^0(g\times h) = \acl^0(g'+h')$
(where $\acl^0(x) = C_0(x)^{\alg}$),
then there are $n \in \N_{>0}$ and an isogeny $\alpha : G \rightarrow G'$
and a point $c \in G'(C_0)$ such that $\alpha g = ng' + c$.
\end{fact}
\begin{proof}
This is a consequence of \cite[Lemme~2.4]{BMP-beauxGroupes}.
Indeed, that lemma yields, via Remark~\ref{rmk:corrGeneric}, that there is
an algebraic subgroup $S \leq G \times G'$ such that the projections to $G$
and $G'$ are surjective and have finite kernels. It follows that $G$ is
abelian.
Indeed, if $g \in G$ then $g^G$ is finite, so the centraliser $C_g$ is a
finite index subgroup of $G$ and hence is equal to $G$ since the latter is
connected. Alternatively, one may assume by the Lefschetz principle that
$G$, $G'$, and $S$ are complex algebraic groups, and observe that $S$
induces an isomorphism of the Lie algebras.
Now let $n$ be the exponent of the subgroup $\operatorname{coker}(S) := \{ y
\in G' : (0,y) \in S \} \leq G'$.
Then by setting $\alpha(x) := ny$ whenever $(x,y) \in S$, we obtain a
well-defined isogeny $\alpha : G \twoheadrightarrow G'$.
So $G$ is isogenous to $G'$.
For the ``moreover'' clause, we use that the subgroup $S$ in the above cited
lemma is a coset of $V := \locus^{G\times G'}((g,g')/C_0)$. Knowing that $G$
is abelian, we can see this fairly directly as follows.
Let $S$ be the stabiliser of $\tp^0(g,g')$, namely $S := \{ \gamma \in
G\times G' : \dim(V \cap (\gamma+V)) = \dim(V) \}$.
Then $S$ projects surjectively with finite kernel to $G$ and to $G'$, and it
follows from our assumptions that $V$ is a coset of $S$;
indeed, this can be seen by applying \cite[Theorem~1]{Ziegler-noteGeneric}
to $(g,g') + (h,h') = (g \times h,g'+h')$.
Since $C_0$ is algebraically closed and both $V$ and $S$ are over $C_0$,
there is $(c_1,c_2) \in (G \times G')(C_0)$ such that $V = (c_1,c_2)+S$.
Then since the projection $\pi_1 : S \twoheadrightarrow G$ is surjective,
there exists $c' \in G'(C_0)$ such that $(g,g'+c') \in S$, namely any $c'$
such that $(c_1,c_2-c') \in S$.
Then with $\alpha,n$ as above, we have $\alpha(g) = n(g'+c') = ng' + nc'$,
so $c := nc'$ is as required.
\end{proof}
\end{parag}
\subsection{Incidence bounds and Szemer\'edi-Trotter}\label{subsec:incidence}
As in \cite{ES-groups} we will require some incidence bounds \`a la
Szemer\'edi-Trotter in higher dimension. As is well-known, if $\mathcal{G}$ is
a bi-partite graph between vertex sets $X_1$ and $X_2$ with the property that
no two distinct points in $X_2$ have more than $B$ common neighbours, then a
simple argument via the Cauchy-Schwarz inequality (e.g.\ see \cite[Prop.
12]{ES-groups}), implies that the number of edges of $\mathcal{G}$ is at most
$O(|X_1|^{\frac{1}{2}}|X_2| + B|X_1|)$. The theorem of Szemer\'edi-Trotter and
its generalisations (such as \cite{pach-sharir}, \cite[Theorem 9]{ES-groups}
or more recently \cite[Theorem~1.2]{FoxPachEtAl}) aim at improving this
inequality by some power saving in the situation when the vertex sets $X_1$
and $X_2$ are points in Euclidean space and the graph $\mathcal{G}$ is given
by some algebraic relation. For example Elekes-Szab\'o prove the following
Szemer\'edi-Trotter-type result:
\begin{theorem}[{{\cite[Theorem 9]{ES-groups}}}]\label{st} If $V \subset
\C^{n_1}
\times \C^{n_2}$ is a complex algebraic subvariety there is
$\epsilon_0=\epsilon_0(n_2)>0$ such that the following holds. Let $B
\in \N$. Let $X_1,X_2 \subset \C^n$ be finite subsets. Write $V(y):=\{x \in
\C^n : (x,y) \in V\}$ for the fiber above $y \in \C^n$. Assume that for any
two distinct $y,y' \in X_2$ the intersection $V(y) \cap V(y')$ contains at
most $B$ points from $X_1$. Then the number $I$ of incidences $(x,y) \in X_1
\times X_2$ with $x \in V(y)$ satisfies:
$$I \leq O_{B,V,n_1}(|X_1|^{\frac{1}{2}(1+\epsilon_0)} |X_2|^{1-\epsilon_0}
+|X_1| + |X_2|\log|X_1|).$$
\end{theorem}
We note that this bound has been slightly improved, with a better $\epsilon_0$
(namely any $\epsilon_0<\frac{1}{4n_2-1}$) and no $\log$ factor in
\cite[Theorem~1.2]{FoxPachEtAl}.
Looking carefully at the proofs of the above theorem we find that the
dependence in $B$ of the big-$O$ is sublinear, that is $O_{B,V,n_1} \leq B \cdot
O_{V,n_1}$ (see \cite[Problem~11.4]{Sheffer-Incidence}). This aspect will be
important for us (we can afford a polynomial dependence).
In what follows we spell out how the above incidence bound reads in the
formalism of coarse pseudo-finite dimension. With the notation and terminology
of Section~\ref{subsect:psfDim} (in particular $K$ is an ultraproduct of
fields of characteristic zero and $\bdl$ is the coarse dimension
\ref{basic-prop}), we have:
\begin{lemma}[Szemer\'edi-Trotter-type bound] \label{lem:SzT}
Let $X_1 \subset K^{n_1}$ and $X_2 \subset K^{n_2}$, suppose each $X_i$ is
$\bigwedge$-internal, and let $X = (X_1 \times X_2) \cap V$ where $V \subset
K^{n_1+n_2}$ is a $K$-Zariski closed subset. Assume that $\bdl(X_1),\bdl(X_2)$
are both finite.
Set
\[ \beta:= \sup_{a,b \in X_2 ;\; a \neq b} \bdl(X(a) \cap X(b)),\]
where $X(y) := \{ x \in X_1 : (x,y) \in X \}$. Then for some $\epsilon_0>0$
depending only on $n_2$, writing $y^+:=\max\{0,y\}$,
\[ \bdl(X) \leq \beta + \max \left(\frac{1}{2}\bdl(X_1) + \bdl(X_2) -
\epsilon_0 (\bdl(X_2) - \frac{1}{2}\bdl(X_1))^{+},\bdl(X_1), \bdl(X_2)
\right) .\]
\end{lemma}
\begin{remark} In the same way, the trivial bound mentioned earlier (via
Cauchy-Schwarz) yields the same estimate on $\bdl(X)$ as above, but with
$\epsilon_0=0$. The original Szemer\'edi-Trotter theorem
\cite{szemeredi-trotter} corresponds to the case when $X_1$ is
the ultraproduct of finite sets of points in $\R^2$ and $X_2$ is the
ultraproduct of finite sets of lines in $\R^2$, and $V$ is the incidence
relation $p \in \ell$. In this case $\epsilon_0=\frac{1}{3}$, which is
optimal.
\end{remark}
\begin{proof}[Proof of Lemma~\ref{lem:SzT}]
Suppose first that $X_1$ and $X_2$ are internal sets, i.e.\ $X_i = \prod_{s
\rightarrow \U} X_i^{K_s}$ for $i=1,2$, for some $X_i^{K_s} \subset
K_s^{n_i}$, and $X^{K_s} = (X_1^{K_s} \times X_2^{K_s}) \cap V(K_s)$. Since
$\bdl(X_i)$ is finite, $X_i^{K_s}$ is finite for $\mathcal{U}$-almost every
$s$. The assumption $\bdl(X(a) \cap X(b))\leq \beta$ for each $a,b \in X_2$
implies that for each $\epsilon>0$ for $\mathcal{U}$-almost every $s$ we
have: $$|X^{K_s}(a) \cap X^{K_s}(b)| \leq B_s,$$ where
$B_s:=\xi_s^{\beta+\epsilon}$, and $\xi=\lim_{s \to \mathcal{U}} \xi_s$ is
the scaling constant as in \S \ref{basic-prop}. Now Theorem \ref{st}
implies:
\[ |X^{K_s}|
\leq B_s \cdot O_{V,n_1}
\left({|X_1^{K_s}|^{\frac{1}{2}(1+\epsilon_0)}|X_2^{K_s}|^{1-\epsilon_0}
+ |X_1^{K_s}| + |X_2^{K_s}| \log |X_1^{K_s}|}\right) . \]
Taking logarithms and passing to the ultralimit yields the desired bound.
Finally the following claim allows us to reduce to the case when $X_i$ are
internal sets:
\begin{claim}
For any $\beta' > \beta$,
there are internal subsets $X_i' \supset X_i$, for $i=1,2$, such that
for all $a,b \in X_2'$ with $a \neq b$, we have $\bdl(X'(a) \cap X'(b)) <
\beta'$, where $X' := (X_1' \times X_2') \cap
V$.
\end{claim}
\begin{proof}
The variety $V$ is defined over a countable (finitely generated) subfield
of $K$, which we denote by $k$.
Since each $X_i$ is $\bigwedge$-internal, we may work in a language $\L$
in which each $X_i$ is $\bigwedge$-definable and $\bdl$ is continuous.
Note that $X(y)=X_1 \cap V(y)$ for each $y \in X_2$. Since $X_1$ is
$\bigwedge$-definable, in view of $(\ref{typedefdel})$, for any $a,b \in
X_2$ with $a\neq b$ there is a $\emptyset$-definable subset $X_1^{a,b}
\supset X_1$ such that $\bdl(X_1^{a,b} \cap V(a) \cap V(b)) < \beta'$. By
continuity of $\bdl$ (see \ref{continuity} and Remark \ref{C-def-cont}),
there is a $k$-definable subset of $Z^{a,b}$ of $K^{n_2}\times K^{n_2}$
containing $(a,b)$ such that $\bdl(X_1^{a,b} \cap V(a') \cap V(b')) <
\beta'$ for all $(a',b') \in Z^{a,b}$. Hence $(X_2)^2 \setminus \Delta$
(where $\Delta$ denotes the diagonal) is covered by the family of
$k$-definable sets $Z^{a,b}$. This is a countable family, because there
are only countably many $k$-definable sets. Combined with the fact that
$X_2$ is $\bigwedge$-definable, $\aleph_1$-compactness (see \ref{sat}) now
implies that there must be a $\emptyset$-definable set $X'_2$ containing
$X_2$ such that $(X'_2)^2 \setminus \Delta$ is contained in finitely many
$Z^{a,b}$'s, say $Z^{a_1,b_1},\ldots,Z^{a_m,b_m}$. Let $X_1'$ be the
intersection of the corresponding $X_1^{a_i,b_i}$, $i=1,\ldots,m$. Then by
monotonicity of coarse dimension $\bdl(X_1' \cap V(a') \cap V(b')) <
\beta'$ for all $a',b' \in (X'_2)^2 \setminus \Delta$. So $X'_1$ and
$X'_2$ are as desired.
\end{proof}
\end{proof}
\section{Warm-up: the Elekes-Szab\'o theorem}\label{sec:warmup}
In this section, we show how the proof of the original Elekes-Szab\'o theorem
translates in the non-standard setup expounded in the previous section. This
will help us motivate the notions introduced in the following section, where
we will pass to the general case of arbitrary dimension and arity and work
towards Theorems \ref{thm:main1} and \ref{thm:main}. We begin with the
one-dimensional case, i.e.\ we prove Theorem \ref{thm:ES}. A similar result
was proven by Hrushovski using similar techniques as
\cite[Proposition~5.21]{Hr-psfDims}. We then proceed to recover
Elekes-Szab\'o's second theorem, which corresponds to the case of a
$2d$-dimensional variety in $(\C^d)^3$, and at the same time add two things:
we establish that the associated algebraic group is in fact commutative (this
was noted already in \cite{breuillard-wang}), and we also give an explicit gap
in the power-saving, $\frac{1}{16}$ in fact. Although this is indeed new, we
include this section mostly for the reader's convenience as a way to introduce
some of the ideas in a special case. But a reader only interested in the proof
of Theorems \ref{thm:main1} and \ref{thm:main} may safely skip ahead to
Section~\ref{sec:proj}.
\begin{parag}{\bf Abelian group configuration theorem.} \label{fact:abGrpConf}
While Elekes-Szab\'o used their `composition lemma' to establish the existence
of the associated algebraic group, we will rely directly on the Group
Configuration Theorem. This is a by now classical theorem of model theory due
to Zilber and Hrushovski. We first recall its statement in the form we need
and then describe a variant, due to Hrushovski, which ensures that the
associated group is commutative. In this paragraph $C_0 \leq K$ are arbitrary
algebraically closed fields, and we use the notation of \S
\ref{notn:baseField}, in particular $K^{<\infty} = \cup_{n>0} K^n$ and
$\acl^0(A)$ is the algebraic closure of $C_0(A)$ in $K$.
\begin{theorem}[Group Configuration Theorem]
\label{thm:groupConf}
Suppose $a,b,c,x,y,z \in K^{<\infty}$ are such that in the following
diagram
\[ \xymatrix{
& & &&&& c \ar@{-}'[ddll][dddlll] \ar@{-}'[dlll][ddllllll] \\
& & & b & & & \\
a & & & & z & & \\
& & & x && & \\
&& & & & & y\;\; \ar@{-}'[uull][uuulll] \ar@{-}'[ulll][uullllll] } \]
for any three distinct points $a_1,a_2,a_3$,
\begin{itemize}\item if $a_1,a_2,a_3$ lie on a common line then
$a_i \in \acl^0(a_j,a_k)$ whenever $\{i,j,k\}=\{1,2,3\}$,
\item if $a_1,a_2,a_3$ do not lie on a common line then $a_i \ind^0 a_ja_k$
whenever $\{i,j,k\}=\{1,2,3\}$.
\end{itemize}
Then there is a connected algebraic group $(G,\cdot)$ defined
over $C_0$, and generic elements $a',b',c'\in G(K)$ such that each primed
element is
$\acl^0$-interalgebraic with the corresponding unprimed element, namely
$\acl^0(x) = \acl^0(x')$ for each $x \in \{a,b,c\}$, and $c'=b'\cdot a'$.
% , $y'=a'+x'$ and $z'=b'+y' = c'+x'$.
\end{theorem}
\begin{remark}
Here, $\acl^0(x')$ is to be understood via a coding of elements of $G(K)$ as
tuples from $K$, as discussed in \ref{absVars}. But since $x'$ is generic,
we may equivalently fix a single arbitrary affine patch over $C_0$ and take
co-ordinates there.
\end{remark}
%\begin{proof}
For a proof of this theorem, we refer the reader to \cite[Theorem~5.4.5,
Remark~5.4.10]{Pillay-GST}. Strictly speaking only a $\bigwedge$-definable
(in ACF) group $G'$ satisfying desired conclusions is obtained there, but by
\cite[Remark~1.6.21]{Pillay-GST}, $G'$ is in fact definable (in ACF). By the
Van den Dries-Hrushovski-Weil theorem \cite[Theorem~4.12]{Pillay-ACF} any
such group is definably isomorphic over $C_0$ to an algebraic group $G$ as
required.
% Justification that we can work over an arbitrary model (commented out in
% favour of pointing to GST 5.4.10)
% \footnote{
% This can be seen as follows. Working first over an $\aleph_1$-saturated
% model, we obtain a $k_\G$-dimensional commutative algebraic group
% $G_{\xitup}$ over a field extension $C_0(\xitup)$ algebraically independent
% from $C_0(abc)$ over $C_0$, and generics $a',b',c':=a'+b'$ with
% interalgebraicity over $C_0(\xitup)$ of the primed points with the
% corresponding unprimed points.
% But now by the coheir property of independence,
% $\tp^{ACF}(\xitup/C_0(abc))$ is finitely satisfiable in $C_0$; using this,
% or alternatively applying an appropriate field specialisation $C_0(\xitup)
% \rightarrow C_0$, we can find $\xitup' \in C_0^{< \infty}$ such that the
% corresponding definable set $G := G_{\xitup'}$ over $C_0(\xitup')=C_0$ is
% still a commutative group of dimension $k_\G$, and $G$ contains points
% $a'',b'',c'':=a''+b''$ such that the interalgebraicities with $a,b,c$ now
% hold over $C_0$, and so $a'',b'',c'' \in G(K)$ are generics of $G$.
% See
% \cite[Lemma~1.2]{EH-autGeomACF} and
% \cite[Lemma~2.1.1,~Theorem~2.1.2]{EH-projACF} for a slightly more direct
% approach.}
% {\bf Add here references to Pillay, Buehler, and Hrushovski's thesis and
% Hrushovski-Zilber's Zarsiki Geometries paper, perhaps also the original
% Zilber reference and Martin's MMM notes ?}
% \mbcomment{I'm not sure about adding further references. Would they
% actually be helpful? Buechler might be, but I'm not so comfortable pointing
% to that as I haven't checked his proof. My notes potentially could be too,
% but really they just repeat Pillay so it seems redundant.}
%\end{proof}
\begin{theorem}[Abelian Group Configuration Theorem]
\label{thm:abGroupConf}
Suppose $a,b,c,w,x,y,z \in K^{<\infty}$ are such that in the following
diagram
\[ \xymatrix{
& & &&&& c \ar@{-}'[ddll][dddlll] \ar@{-}'[dlll][ddllllll] \\
& & & b \ar@{-}'[d][dd] & & & \\
a & & & w & z \ar@{-}'[l][llll] & & \\
& & & x && & \\
&& & & & & y\;\; , \ar@{-}'[uull][uuulll] \ar@{-}'[ulll][uullllll] } \]
for any three distinct points $a_1,a_2,a_3$,
\begin{itemize}\item if $a_1,a_2,a_3$ lie on a common line then
$a_i \in \acl^0(a_j,a_k)$ whenever $\{i,j,k\}=\{1,2,3\}$,
\item if $a_1,a_2,a_3$ do not lie on a common line and $\{a_1,a_2,a_3\} \neq
\{w,c,y\}$ then $a_i \ind^0 a_ja_k$ whenever $\{i,j,k\}=\{1,2,3\}$.
\end{itemize}
Then there is an connected commutative algebraic group $G$ defined
over $C_0$, and generics $a',b',c'\in G(K)$ such that each primed element is
$\acl^0$-interalgebraic with the corresponding unprimed element, and
$c'=b'+a'$.
% , $y'=a'+x'$ and $z'=b'+y' = c'+x'$.
\end{theorem}
%\begin{proof}
Note that the hypotheses of Theorem~\ref{thm:groupConf} are satisfied, so we
need only show that our additional assumptions yield that the algebraic group
$G$ obtained from that theorem is commutative. We refer to
\cite[Theorem~C.1]{BHM-CCMA} for a proof of this.
% \mbcomment{I don't think it makes sense to try to write a friendlier
% version of that proof to go here, since a reader who can't read the
% original proof will also have a hard time with the proofs of the group
% configuration theorem. I don't see that it can be made much friendlier,
% anyway.}
%\end{proof}
\end{parag}
\begin{parag}{\bf Elekes-Szab\'o - one dimensional case.} In this paragraph we
reprove the original Elekes-Szab\'o theorem, namely Theorem \ref{thm:ES}. We
start by reformulating it in the non-standard setup of the last section; in
particular we keep the notation of Section~\ref{subsect:psfDim}. So $K$ is
an ultrapower of the complex field, $\bdl$ is the coarse dimension
\ref{basic-prop} which is continuous in a countable language $\L$ containing
$\L_{ring}$ and constant symbols for each element of the countable
algebraically closed field $C_0$ over which $V$ is defined, and $\dimo$
denotes transcendence degree over $C_0$.
\begin{theorem}[Reformulation of Theorem \ref{thm:ES}]\label{thm:ES-ref} Let
$a_1, a_2, a_3 \in K$ and assume that for all $i\neq j$,
$dim^0(a_i,a_j ) = dim^0(a_1,a_2,a_3) = 2$, $\bdl(a_i) \leq 1$ and
$\bdl(a_1,a_2,a_3) = 2$.
Then there exists a connected one-dimensional algebraic group $G$ over $C_0$ and
$a'_1,a'_2,a'_3 \in G(K)$ with $\acl^0(a_i)=\acl^0(a'_i)$ for $i=1,2,3$ and
$a'_3=a'_1+a'_2$.
\end{theorem}
\begin{proof}[Proof of Theorem \ref{thm:ES} from Theorem \ref{thm:ES-ref}]
Assume $V \subset \C^3$ does not project to a curve on two co-ordinates and
has no power-saving. Then we may find a sequence of positive integers
$(N_s)_{s \ge 0}$ with $\lim_{s \to \infty} N_s=+\infty$ and finite subsets
$X^s_1,X^s_2$ and $X^s_3$ in $\C$ with $|X^s_i| \leq N_s$ for each $i,s$
such that $$|X^s_1 \times X^s_2 \times X^s_3 \cap V| \geq
N_s^{2-\epsilon_s}$$ for some $\epsilon_s>0$ with $\lim_{s \to \infty}
\epsilon_s=0$. Passing to an ultraproduct $X_i=\prod_{s \to
\mathcal{U}}X^s_i$ for some non-principal ultrafilter $\mathcal{U}$ over the
integers, we obtain three internal sets $X_i \subset K$, where $K$ is the
ultrapower of $\C$, and we define the coarse dimension $\bdl$ as in
\ref{basic-prop} with scaling constant $\xi=\lim_{s \to \mathcal{U}} N_s$.
Hence $\bdl(X_i) \leq 1$ for each $i$ and $\bdl(X_1 \times X_2 \times X_3
\cap V) =2$.
Since $V$ is irreducible and does not project to a curve on two co-ordinates,
the fibers of co-ordinate projections of $V$ on pairs of co-ordinates have
uniformly bounded size. Consequently $|X^s_1 \times X^s_2 \times X^s_3 \cap V|
= O(|X^s_i \times X^s_j|)$ for all $s$ and all $i \neq j$. It follows that
$2=\bdl(X_1 \times X_2 \times X_3 \cap V) \leq \bdl(X_i) +\bdl(X_j)$, and
hence that $\bdl(X_i)=1$ for each $i$.
The variety $V$ is defined over some finitely generated subfield of $\C$. Let
$C_0$ be its algebraic closure in $\C$. It is a countable subfield. To be able
to talk about definable sets we specify a language $\L$ as follows: we start
with $\L_{ring}=(K,+,\cdot,0,1)$ the language of rings and enlarge it by
adding a constant symbol for each element of $C_0$ as well as a predicate for
each $X_i$, $i=1,2,3$, thus in effect forcing $X_i$ to be definable. Finally
we enlarge $\L$ as in \S \ref{continuity} so as to make $\bdl$ continuous and
hence additive.
Now by Fact \ref{fact:ideal} we may find a triple $(a_1,a_2,a_3) \in X_1
\times X_2 \times X_3 \cap V$ such that $\bdl(a_1,a_2,a_3) = \bdl(X_1 \times
X_2 \times X_3 \cap V) =2$. Note that $(a_1,a_2,a_3)$ is generic in $V$,
i.e.\ it is not contained in any proper algebraic subvariety over the base
field $C_0$, because $|X^s_1 \times X^s_2 \times X^s_3 \cap W| =O_W(N_s)$ for
every one-dimensional subvariety $W \subsetneq V$ over $C_0$, and so $\bdl(X_1
\times X_2 \times X_3 \cap W) \leq 1$. Consequently
$\dimo(a_1,a_2,a_3)=\dimo(a_i,a_j)=2$ for all $i\neq j$. So we are in the
situation of Theorem \ref{thm:ES-ref}. Then $\loc^0(a'_1,a'_2,a'_3)$ is the
graph $\Gamma_G(\C)$ of the group operation of $G$, and we conclude that $V$
has the required description via the correspondence
$\loc^0((a_1,a_2,a_3),(a'_1,a'_2,a'_3)) \subset V \times \Gamma_G(\C)$, which
is defined over $C_0$ and projects to the correspondences given by the
(irreducible) curves $\loc^0(a_i,a'_i) \subset \C \times G(\C)$.
\end{proof}
We now pass to the proof of Theorem \ref{thm:ES-ref}. We need to verify that
the hypotheses of the group configuration are met. For this we crucially need
the following lemma, which can be interpreted as saying that a $2$-parameter
family of plane curves with no power-saving must in fact be one-dimensional.
This is where the Szemer\'edi-Trotter bound comes into play.
\begin{lemma}\label{1base} Let $x_1,\ldots,x_4 \in K$ be such that $\bdl(x_i)=
1$ and $\bdl(x_1,\ldots,x_4)=\dimo(x_1,\ldots,x_4)$. Assume that
$\dimo(x_1,x_2/x_3,x_4)=1$. Then there is $x_5 \in \acl^0(x_3,x_4)$ with
$\bdl(x_5)=\dimo(x_5)=1$ such that $\dimo(x_1,x_2/x_5)=1$.
\end{lemma}
\begin{proof} We postpone the proof of this lemma to the next subsection,
where a stronger quantitative version of it will be proven as Lemma
\ref{1base-m}. It is also a special case of Proposition
\ref{lem:cgpCoherentLinearity}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:ES-ref}] First note that the
assumptions imply that $\bdl(a_i)=1$ for each $i$. Indeed for any three
distinct $i,j,k$ we have $a_i \in \acl^0(a_j,a_k)$. Hence by
$(\ref{alg-ineq})$ we have $\bdl(a_i/a_j,a_k)=0$. And by additivity of
$\bdl$ (see Fact \ref{fact:dlContProps}) we get
\begin{equation}\label{delcal} \bdl(a_i,a_j,a_k) = \bdl(a_i/a_j,a_k) +
\bdl(a_j,a_k) = \bdl(a_j,a_k) \leq \bdl(a_j)+\bdl(a_k).
\end{equation} This forces $\bdl(a_j)$ and $\bdl(a_k)$ to be equal to $1$,
since both are $\leq 1$.
Let $X=\tp(a_2,a_3/a_1)(K)$ be the set of realisations of the type of the pair
$(a_2,a_3)$ over $a_1$, namely the intersection of all definable sets over
$C:=\{a_1\}$ containing $(a_2,a_3)$. By additivity of $\bdl$ we have
$\bdl(X)=\bdl(a_1,a_2,a_3)-\bdl(a_1)=2-1=1$, by assumption. According to Fact
\ref{fact:ideal} we can find $(a_4,a_5) \in X$ such that
\begin{equation}\label{bdlX}\bdl(a_4,a_5/a_1,a_2,a_3) =
\bdl(X)=1.
\end{equation}
We will show that there is $a_6 \in K$ such that $a_1,\ldots,a_6$ satisfy the
hypotheses of the group configuration theorem as in the following diagram:
\[ \xymatrix{
& & &&&& a_3 \ar@{-}'[ddll][dddlll] \ar@{-}'[dlll][ddllllll] \\
& & & a_2 & & & \\
a_1 & & & & a_6 & & \\
& & & a_4 && & \\
&& & & & & a_5\;\; \ar@{-}'[uull][uuulll] \ar@{-}'[ulll][uullllll] } \]
Since $(a_4,a_5)$ and $(a_2,a_3)$ have the same type over $a_1$, they have the
same type over the empty set, and in particular they belong to the same
algebraic subsets of $K^2$ defined over $C_0$. So $\dimo(a_4,a_5,a_1)=2$,
$\dimo(a_4,a_5)=\dimo(a_4,a_1)=\dimo(a_5,a_1)=2$.
Moreover the Zariski dimension of the whole system is $3$, i.e.\
$\dimo(a_1,\ldots,a_5)=3$. Indeed it is at most $3$ given that $a_5 \in
\acl^0(a_1,a_4)$ and $a_3 \in \acl^0(a_1,a_2)$, but it cannot be less, for
otherwise $\dimo(a_4,a_5/a_1,a_2,a_3)=0$ forcing $\bdl(a_4,a_5/a_1,a_2,a_3)=0$
by $(\ref{alg-ineq})$, a contradiction to $(\ref{bdlX})$.
$$ \emph{Claim:} \hspace{.5cm} \dimo(a_3,a_4)=2, \dimo(a_2,a_5)=2
\textnormal{ and }\dimo(a_2,a_5/a_3,a_4)=1.$$
Indeed if $\dimo(a_3,a_4)<2$, then $a_4 \in \acl^0(a_3)$, and thus
$$\dimo(a_1,\ldots,a_5)= \dimo(a_1,a_2,a_3,a_4)= \dimo(a_1,a_2,a_3)=2,$$ where
we have used that $a_5 \in \acl^0(a_1,a_4)$. In a similar way
$\dimo(a_2,a_5)=2$. Now by additivity of $\dimo$ we finally get
$\dimo(a_2,a_5/a_3,a_4)=1$, proving the claim.
Further note that by additivity and $(\ref{bdlX})$ we have
$$\bdl(a_2,a_3,a_4,a_5)=\bdl(a_4,a_5/a_2,a_3)+\bdl(a_2,a_3)=1+2=3=\dimo(a_2,a_3,a_4,a_5).$$
So Lemma \ref{1base} applies and gives $a_6 \in \acl^0(a_3,a_4)$ such that
$\dimo(a_2,a_5/a_6)=1$ and $\dimo(a_6)=1$. It then follows easily by
additivity of $\dimo$ that $\dimo(a_6,a_2)=\dimo(a_5,a_6)=2$ and $a_6 \in
\acl^0(a_2,a_5)$. This shows that $a_1,\ldots, a_6$ satisfy the hypotheses of
the group configuration theorem. We are done.
\end{proof}
\end{parag}
\begin{parag}{\bf Coarse general position.} \label{cgp-parag}
A significant new difficulty arises when dealing with the higher dimensional
situation, i.e.\ when $m=\dim W_i>1$ say in Theorem \ref{thm:main}. We will
have to assume that the finite sets $X_i \subset W_i$ do not have too large an
intersection with proper subvarieties. There are various ways to quantify this
assumption, for instance Elekes-Szab\'o's notion of \emph{general position}
requires that the intersections have bounded size with a bound depending only
on the complexity of the subvariety. We will adopt here the weaker assumption
of \emph{coarse general position}. As explained in Section \ref{gpNecessity}
below, some assumption of this kind is \emph{necessary} for the result to
hold.
Recall from Definition \ref{defn:taucgp} that for $\tau \in \N$, a finite
subset $X$ of a complex algebraic variety $W$ is said to be in
\defnstyle{coarse $(C,\tau)$-general position} (or $(C,\tau)$-$cgp$ for short)
with respect to $W$ if $|W' \cap X| \leq |X|^{\frac 1\tau}$ for any proper
irreducible complex subvariety $W' \subsetneq W$ of complexity at most $C \in
\N$. In the non-standard setup of Section \ref{sec:setup}, where we have
specified a language $\L$ and defined the coarse dimension $\bdl$, it will be
convenient to define a notion of coarse general position for tuples $\a \in
K^{<\infty}$. We will say that $\a \in K^{<\infty}$ is in coarse general
position or is \defnstyle{cgp} for short if for every $\b \in K^{<\infty}$
such that $\a$ is not independent from $\b$, that is such that $\dimo(\a/\b) <
\dimo(\a)$, we have: $$\bdl(\a/\b)=0.$$
The two notions are closely related as follows. Suppose $W \subset \C^n$ is a
variety and $X = \prod_{s \rightarrow \U} X_s \subset W(K)$ is an internal
set. Assume that $X$ is definable without parameters in the countable language
$\L$ of Section \ref{sec:setup} for which the coarse dimension $\bdl$ is
continuous (see \ref{continuity}).
\begin{lemma}\label{cgplemfin} Suppose that $0 < \bdl(X) < \infty$ and that
for any $\tau \in \N$, there is $C \ge \tau$ such that for $\U$-many $s$,
$X_s$ is $(C,\tau)$-$cgp$ in $W(\C)$. Then any tuple $\a \in K^n$ lying in
$X$ is $cgp$.
\end{lemma}
\begin{proof}Let $\b \in K^{<\infty}$ such that $\dimo(\a/\b) < \dimo(\a)$.
Then setting $W'=\loc^0(\a/\acl^0(\b))$ we get an absolutely irreducible
subvariety of $W$, which is proper, since by $(\ref{dim-loc})$ one has
$\dim(W')=\dimo(\a/\b) < \dim W$, and contains $\a$. Let $c$ be the
complexity of $W'$. Then for every $\tau>c$, $\U$-many $X_s$ are
$(c,\tau)$-$cgp$ in $W(\C)$, and this implies that $\bdl(\a/\acl^0(\b)) \leq
\bdl(X \cap W'(K)) =0$. Hence $\bdl(\a/\b)=0$ by $(\ref{alg-ineq})$.
\end{proof}
\begin{remark}\label{cgp-invariance} The property of being $cgp$ for a tuple
$\a \in K^m$ depends only on the type $\tp(\a)$ of $\a$. Indeed, suppose
$\a'\in K^m$ has the same type, and $\b \in K^n$ is such that $\a' \nind^0
\b$. Then there is $\b' \in K^n$ such that $\tp(\a,\b') = \tp(\a',\b)$, by
$\aleph_1$-compactness of $K$. So $\a \nind^0 \b'$, and so
$\bdl(\a/\b')=0$. But then by invariance of $\bdl$ (see
\ref{fact:dlContProps}ii.), we have $\bdl(\a'/\b)=\bdl(\a/\b)=0$.
\end{remark}
\end{parag}
\begin{parag}{\bf Higher dimensional case: Elekes-Szab\'o with a gap and
commutativity.} We now move on to the second theorem proved by
Elekes-Szab\'o in \cite{ES-groups}, which is the extension of Theorem
\ref{thm:ES} to higher dimensional varieties. We give a proof following the
strategy used above in the one-dimensional case. As a payoff we will also
get an explicit bound, $\frac{1}{16}$, on the power-saving and we will
establish that the group involved must be commutative. This feature (rather
the nilpotency) had been hinted at already by Elekes-Szab\'o (see their
Example 28 in \cite{ES-groups}), but was first established by H. Wang and
the second named author \cite{breuillard-wang} via a different argument
using the classification of approximate groups from \cite{bgt}.
As before we consider three irreducible complex varieties $W_1,W_2,W_3$ of
dimension $d$.
We say that a subvariety $V \subset \prod_i W_i$ admits a power-saving
$\eta>0$ if there exists $\tau \in \N$ such that
$$|V \cap X_1 \times X_2 \times X_3| \leq O_{V,\tau}(N^{\dim(V) - \eta}).$$
for every $N \in \N$ and all finite subsets $X_i \subset W_i$ with $|X_i|\leq
N^d$ and each $X_i$ in coarse $\tau$-general position in $W_i$.
\begin{theorem} \label{cor:ES}
Suppose $V \subset W_1\times W_2 \times W_3$ are irreducible complex
varieties, and $\dim(W_i)=d$ and $\dim(\pi_{ij}(V)) = 2d = \dim(V)$ for all
$i\neq j \in \{1,2,3\}$. Then either $V$ admits a power-saving $\frac{1}{16}$,
or $V$ is in co-ordinatewise correspondence with the graph $\Gamma_+ \subset
G^3$ of the group operation of a commutative algebraic group $G$.
\end{theorem}
\begin{remark} \label{rmk:ESGap2}
Note that we obtain a power saving which is independent of $d$. In fact the
method gives a power-saving $\eta$ for any $\eta<\frac{d}{16d-1}$, which is
slightly better. For $d=1$, a power-saving of $\frac16$ was obtained by Wang
\cite{Wang-gap} and independently by Raz, Sharir, and de Zeeuw \cite{RSZ-ESR}.
The method given below also gives $\frac{1}{6}$ when $d=1$, see Remark
\ref{best-exp}.
\end{remark}
The remainder of this section is devoted to the proof of Theorem \ref{cor:ES}.
As in the one-dimensional case, we first reformulate the result in the
framework of coarse dimension using the notation of Section \ref{sec:setup}.
\begin{theorem}[Reformulation of Theorem \ref{cor:ES}] \label{thm:coherentES}
Suppose $\a_1,\a_2,\a_3 \in K^d$ with $\dimo(\a_i,\a_j ) =
\dimo(\a_1,\a_2,\a_3) = 2d$, $\bdl(\a_i) \leq d$ and $\bdl(\a_1,\a_2,\a_3)
\in [2d - \frac{1}{16},2d]$. Assume that each $\a_i$ is $cgp$ in the sense
of \ref{cgp-parag}.
Then there is an $d$-dimensional commutative algebraic group $G$ over $C_0$,
and $\a'_1,\a'_2,\a'_3 \in G(K)$
such that $\dimo(\a'_i)=d$, $\acl^0(\a_i)=\acl^0(\a'_i)$ and
$\a'_1+\a'_2=\a'_3$.
\end{theorem}
\begin{proof}[Reduction of Theorem \ref{cor:ES} to Theorem
\ref{thm:coherentES}] This is essentially the same argument as in the
one-dimensional case, so we shall be brief. Let $\eta=\frac{1}{16}$. Arguing
by contradiction and carefully negating quantifiers we obtain an increasing
sequence of integers $(N_s)_{s \ge 0}$ and a sequence of finite sets $X_i^s
\subset W_i(\C)$ in coarse $s$-general position with $|X_i^s| \leq N_s^d$
but $|X^s_1 \times X^s_2 \times X^s_3 \cap V| \geq N_s^{2d-\eta}$. Passing
to an ultralimit we obtain three internal sets $X_i \subset W_i(K)$, which
we add as predicates to our language (thus turning them into definable
sets). Clearly $\bdl(X_i)\leq d$ and $\bdl(X_1 \times X_2 \times X_3 \cap V)
\geq 2d-\eta$. By Fact \ref{fact:ideal} we find three tuples $\a_i \in X_i$
such that $\bdl(\a_1,\a_2,\a_3) = \bdl(X_1 \times X_2 \times X_3 \cap V)$.
By Lemma \ref{cgplemfin} each $\a_i$ is $cgp$. Moreover $(\a_1,\a_2,\a_3)$
is generic in $V$, for otherwise it would be contained in a subvariety $W
\subsetneq V$ over $C_0$ forcing $\bdl(\a_1,\a_2,\a_3) \leq \bdl(X_1 \times
X_2 \times X_3 \cap W) \leq \dim W\leq2d-1$ (see Lemma
\ref{lem:power-saving}). Therefore $\dimo(\a_i,\a_j ) =
\dimo(\a_1,\a_2,\a_3) = 2d$ and the assumptions of Theorem
\ref{thm:coherentES} are met.
\end{proof}
Analogously to the one-dimensional case, the Szemer\'edi-Trotter bounds of
Lemma \ref{lem:SzT} will be used to prove the following crucial step, which
shows that the $2$-parameter family of varieties $\loc^0(\a/\b)$ is in fact a
$1$-parameter family.
\begin{lemma}\label{1base-m} Let $\eta \in [0,\frac{d}{8d-1})$. Let
$\x_1,\ldots,\x_4 \in K^d$, and set $\a=(\x_1,\x_2)$ and $\b=(\x_3,\x_4)$.
Assume each $\x_i$ is $cgp$ and $\bdl(\x_i)\leq \dimo(\x_i)=d$. Assume
further $\bdl(\a,\b)\geq \dimo(\a,\b) - \eta$ and $\dimo(\a)=\dimo(\b)=2d$
and $\dimo(\a/\b)=d$. Then there is $\x_5 \in \acl^0(\b) ^{<\infty} \cap
\acl^0(\a)^{<\infty}$ with $\dimo(\x_5)=\dimo(\a/\x_5)=d$.
\end{lemma}
\begin{proof} First note that $\bdl(\b) \geq \dimo(\b) - \eta$. To see this
observe that by $cgp$ either $\bdl(\x_2/\b\x_1) = 0$ or $\bdl(\x_1/\b) = 0$.
Indeed otherwise $\dimo(\x_2/\b\x_1)=\dimo(\x_2)=d$ and
$\dimo(\x_1/\b)=\dimo(\x_1)=d$, contradicting $\dimo(\x_1,\x_2/\b)=d$. So we
have $\bdl(\a/\b) = \bdl(\x_2/\b\x_1) + \bdl(\x_1/\b) \leq d =
\dimo(\a/\b)$. And so $\bdl(\b) = \bdl(\a,\b) - \bdl(\a/\b)\ge 3d-\eta -d
=\dimo(\b) - \eta$.
Let $\d \in K^{<\infty}$ be a canonical base for $\a$ over $\b$ (see \S
\ref{parag:canonical}). By definition $\d \in \acl^0(\b)^{< \infty}$. We will
show that this $\x_5:=\d$ satisfies the desired conclusion. By definition we
have $\dimo(\a/\d)=\dimo(\a/\b)=d$. Hence
$\dimo(\d)=\dimo(\a,\d)-\dimo(\a/\d)\ge \dimo(\a)-d=d$.
It only remains to show the upper bound $\dimo(\d) \leq d$. Indeed this will
also imply that $\d \in \acl^0(\a)^{<\infty}$, because then $\dimo(\d)=d$ and
$\dimo(\a\d)=\dimo(\a/\d)+\dimo(\d)=2d$, so $\dimo(\a\d)=\dimo(\a)$ and thus
$\dimo(\d/\a)=0$.
So suppose that $\dimo(\d) > d$. Then $\dimo(\b/\d)=\dimo(\b) -\dimo(\d)0$.
So we must conclude that
$$\bdl(\a/\b) + \bdl(\d) \leq \bdl(X) \leq \frac{1}{2}\bdl(\a) + \bdl(\d) -
\epsilon_0 \big(\bdl(\d) - \frac{1}{2}\bdl(\a)\big).$$
In other words, since $\bdl(\d)=\bdl(\b)$:
$$\frac{1}{\epsilon_0}\bdl(\a\b) \leq \frac{1}{2}
\bdl(\a)(\frac{1}{\epsilon_0} +1) + \bdl(\b)( \frac{1}{\epsilon_0} -1).$$
Finally since $\bdl(\a), \bdl(\b) \leq 2d$ and $\bdl(\a\b) \geq 3d - \eta$ by
assumption, we conclude that
$$\eta \geq \frac{d}{8d-1}$$
a contradiction to our assumption.
We assumed above that $\dimo(\d) < 2d$, and so we conclude from this
contradiction that $\dimo(\d) = d$ or $\dimo(\d) = 2d$. it remains to rule out
the latter case. If we were willing to weaken our bound to
$\frac{d}{4(2d+1)-1}$, the argument above would suffice. To obtain the sharper
bound, we use an additional argument inspired by \cite[Theorem~4.3]{RSZ-ESR}.
So assume $d^0(\d) = 2d$, i.e.\ $\acl^0(\d)=\acl^0(\b)$.
Let $W := \loc^0(\a\b)$
and $V := \loc^0(\a\d)$.
Write $V_{\y}$ for $\{ \x : (\x,\y) \in V \}$, and similarly for $W_{\y}$.
So $V_{\d} = \loc^0(\a/\d)$ is the irreducible component of $W_{\b} =
\loc^0(\a/\b)$ containing $\a$.
Now $\tp^0(\d/\b)(K)$ is finite, so say $\tp^0(\d/\b)(K) = \{\d_1,\ldots
,\d_k\}$.
Then $V_{\d_i}$ are precisely the irreducible components of $W_{\b}$, and
$W_{\b} = \bigcup_i V_{\d_i}$.
Indeed, any $V_{\d_i}$ is an automorphic image over $C_0(\b)$ of $V_{\d}$ and
so is a component, and conversely $\bigcup_i V_{\d_i}$ is
automorphism-invariant over $C_0(\b)$ and hence is defined over $C_0(\b)$,
so $W_{\b} \subset \bigcup_i V_{\d_i}$.
\begin{claim} \label{clm:finDegree}
There are only finitely many $\b' \in \tp^0(\b)(K)$ with
$\dim(W_{\b} \cap W_{\b'}) = d$.
\end{claim}
\begin{proof}
Fix field automorphisms $\tau_{ij} \in \Aut(K/C_0)$
such that $\tau_{ij}(\d_i) = \d_j$. Suppose $\b' \in \tp^0(\b)(K)$ and
$\operatorname{dim}(W_{\b} \cap W_{\b'}) = d$.
Then $W_{\b}$ and $W_{\b'}$ share a component,
so say $V_{\d_j} \subset W_{\b'}$. Now let $\sigma \in \Aut(K/C_0)$ be such
that $\b' = \sigma(\b)$.
Since the $\sigma(V_{\d_i})$ are the irreducible components of $W_{\b'}$,
for some $i$ we have $\sigma(V_{\d_i}) = V_{\d_j}$.
Let $\sigma' := \tau_{ij}^{-1} \sigma$.
Then $\sigma'(V_{\d_i}) = \tau_{ij}^{-1} V_{\d_j} = V_{\tau_{ij}^{-1} \d_j}
= V_{\d_i}$,
so by the definition of canonical base,
we have $\sigma'(\d_i) = \d_i$. Then since $\b \in \acl^0(\d_i)$,
there are only finitely many possibilities for $\sigma'(\b)$,
and so there are only finitely many possibilities for
$\b' = \sigma(\b) = \tau_{ij}(\sigma'(\b))$.
\end{proof}
Now let $G$ be the graph with vertex set $\tp(\b)(K)$ and with an edge between
$\b'$ and $\b''$ if and only if $\dim(W_{\b'} \cap W_{\b''}) = d$. By
Claim~\ref{clm:finDegree}, $G$ has constant finite degree.
\begin{claim}\label{clm:bigAnticlique}
If $G=(A,E)$ is a graph where the vertex set $A$ is $\bigwedge$-internal and
the edge relation $E$ is internal, and if $G$ has finite maximal degree $k$,
then there is a $\bigwedge$-internal anticlique $A'$ with
$\bdl(A')=\bdl(A)$.
% Note actually we get the same fine dimension, or alternatively that we can
% take $d$ pseudo-finite with ``$\bdl(d)=0$''.
\end{claim}
\begin{proof}
If $A$ is internal, then $G$ is the ultraproduct of finite graphs $G_i =
(A_i,E_i)$ of maximal degree $k$. Then $G_i$ has chromatic number at most
$k+1$, and so has an anticlique of size at least $ \frac{|A_i|}{k+1}$.
The ultraproduct of such anticliques is then an internal anticlique $A'
\subset A$ as required.
In general, our $\bigwedge$-internal $A$ is, by $\aleph_1$-compactness of
the ultraproduct, contained in an internal $A_0$ such that $(A_0,E)$ has
maximal degree at most $k$, because the property of having maximal degree at
most $k$ can be expressed as the inconsistency of a partial $(k+1)$-type. So
then the same holds for all internal $A_1$ with $A \subset A_1 \subset A_0$,
and hence the claim follows from the internal case.
\end{proof}
Now let $X_2 \subset \tp(\b)(K)$ be an anticlique as in
Claim~\ref{clm:bigAnticlique} for the graph defined above, and $X_1 :=
\tp(\a)(K)$ and $X := (X_1 \times X_2) \cap W$. If $\a' \in X(\b_1) \cap
X(\b_2)$ then $\dimo(\a'/\b_1\b_2) \leq \dim(W_{\b_1}\cap W_{\b_2}) < d$ since
$X_2$ is an anticlique, and so $\bdl(\a')=0$ by $cgp$. So we contradict the
Szemer\'edi-Trotter bound exactly as in the case $d<\dimo(\d)<2d$ above.
This contradiction shows that $\dimo(\d)=d$ and ends the proof.
\end{proof}
%As above we postpone the proof of this lemma to the next section, where it
%will be shown in greater generality in Proposition \ref{quant}.
\begin{proof}[Proof of Theorem \ref{thm:coherentES}]Here again the strategy is
the same as in the $1$-dimensional case, so we shall be brief. Let
$\eta=\frac{1}{16}$.
%As before we first note that the assumptions imply that $\bdl(\a_i) \in
%(d-\eta,d]$ for each $i$. Indeed this follows as earlier since for distinct
%$i,j,k$ we have $\a_i \in \acl^0(\a_j,\a_k)$ and $(\ref{delcal})$ holds.
As before set $X=\tp(\a_2,\a_3/\a_1)(K)$ and note that $\bdl(\a_1,\a_2,\a_3) =
\bdl(X)+\bdl(\a_1)$ by additivity of $\bdl$. It follows that $\bdl(X) \geq
d-\eta$.
%On the other hand $\bdl(X)=\bdl(\a_2/\a_3,\a_1)+\bdl(\a_3/\a_1) \leq
%\bdl(\a_3)=d$ because $\bdl(\a_2/\a_3,\a_1)=0$ by $(\ref{alg-ineq})$ since
%$\a_2 \in \acl^0(\a_1,\a_3)$.
By Fact \ref{fact:ideal} we may find $(\a_4,\a_5) \in X$ with
$\bdl(a_4,a_5/a_1,a_2,a_3) = \bdl(X)$. Note that $\a_4,\a_5$ are both $cgp$
(see Remark \ref{cgp-invariance}). We will show that there are $\a_6$ and
$\a_7$ in $K^d$ such that $(\a_1,\ldots,\a_7)$ satisfy the hypotheses of the
abelian group configuration theorem as in the following diagram:
\[ \xymatrix{
& & &&&& \a_1 \ar@{-}'[ddll][dddlll] \ar@{-}'[dlll][ddllllll] \\
& & & \a_2 \ar@{-}'[d][dd] & & & \\
\a_3 & & & \a_6 & \a_4 \ar@{-}'[l][llll] & & \\
& & & \a_5 && & \\
&& & & & & \a_7\;\; , \ar@{-}'[uull][uuulll] \ar@{-}'[ulll][uullllll] } \]
As earlier we have that $\dimo(\a_4,\a_5,\a_1)=2d$,
$\dimo(\a_4,\a_5)=\dimo(\a_4,\a_1)=\dimo(\a_5,\a_1)=2d$. And we also have
$\dimo(\a_1,\ldots,\a_5)=3d$. Indeed otherwise
$\dimo(\a_4,\a_5/\a_1,\a_2,\a_3)0$.
Again as in the $1$-dimensional case, using only the additivity of $\dimo$ we
conclude that $\dimo(\a_3,\a_4)=\dimo(\a_2,\a_5)=2d$ and
$\dimo(\a_2,\a_5/\a_3,\a_4)=d$, and also that
$\dimo(\a_2,\a_4)=\dimo(\a_3,\a_5)=2d$ and $\dimo(\a_3,\a_5/\a_2,\a_4)=d$.
Moreover $\bdl(\a_2,\a_3)=\bdl(\a_1,\a_2,\a_3)$ by additivity since $\a_1$ is
$cgp$. Similarly by additivity
$$\bdl(\a_2,\a_3,\a_4,\a_5)=\bdl(\a_1,\ldots,\a_5)=\bdl(X) +
\bdl(\a_1,\a_2,\a_3),$$
hence this is $\ge 3d- 2\eta$. Since $2\eta \leq \frac18 < \frac{d}{8d-1}$, we
are thus in a position to apply Lemma \ref{1base-m} to both
$\a_2,\a_5,\a_3,\a_4$ and to $\a_2,\a_4, \a_3,\a_5$. This yields $\a_6$ and
$\a_7$ as desired and concludes the proof of the theorem.
\end{proof}
\begin{remark}[Quality of the power-saving]\label{best-exp} The quality of the
power-saving depends crucially on the quality of $\epsilon_0$ in the
Szemeredi-Trotter type bound of Lemma \ref{lem:SzT}. We immediately lose a
factor $2$ because this bound is usually proven for real algebraic varities,
while we consider complex varieties and somewhat carelessly view them as
real varieties of twice the dimension. It is plausible that the bound
$\epsilon_0=\frac{1}{2n-1}$ holds in Theorem \ref{st}. In fact this is known
when $n=1$ (see \cite{zhal-szabo-sheffer} or \cite[Thm 4.3]{RSZ-ESR}), and
consequently Lemma \ref{1base-m} holds for all $\eta \in [0,\frac13)$ when
$d=1$ and thus yields a power-saving $\eta$ for any $\eta<\frac16$ in the
$1$-dimensional Elekes-Szab\'o theorem. This recovers the bound obtained in
\cite{Wang-gap} and \cite{RSZ-ESR}. The latter work however gave more
precise information on the multiplicative constant and the dependence on the
degree of the variety $V$, which is an aspect we do not investigate in our
paper (it would require working with the Hrushovski-Wagner fine
pseudo-finite dimension while we restrict attention to the coarse dimension
$\bdl$).
\end{remark}
The following corollary indicates the robustness of the commutativity of the
group in Theorem~\ref{cor:ES}.
\begin{corollary} \label{specialAbelian}
Suppose $(G;\cdot)$ is a connected complex algebraic group.
Suppose the graph $\Gamma$ of multiplication admits no power-saving.
Then $G$ is commutative.
\end{corollary}
\begin{proof}
By Theorem~\ref{cor:ES}, $\Gamma$ is in co-ordinatewise correspondence with
the graph $\Gamma_+ \subset G'^3$ of the group operation of a commutative
connected algebraic group $G'$.
So this is an immediate consequence of Fact~\ref{fact:corrIsog}.
\end{proof}
\begin{remark} Another proof of this corollary was noted in
\cite{breuillard-wang}. It can be derived as a consequence of the
Balog-Szemeredi-Gowers-Tao theorem combined with \cite[Theorem 2.5]{bgt},
one of the main results of \cite{bgt} which was proven there for linear
algebraic groups, but can be extended to all algebraic groups.
%Indeed if $A,B,C$ are finite subsets of $G$ of size $\Leq N$ with say $A
%\xtimes B \times C \cap \Gamma$ of size at least $N^{2-\epsilon}$, then the
%multiplicative energy $E(A,B)$ is at least $N^{3-\epsilon}$ and hence the
%Balog-Szemeredi-Gowers-Tao theorem combined with \cite[Theorem 2.5]{bgt}
%implies that $A$ is contained in fewer than $|A|^{O_G(\epsilon)}$ cosets of
%some nilpotent subgroup of $G$. Some coset of its center will have a large
%intersection with $A$. Since $A$ is assumed to be in coarse general
%position, this forces $G$ to be commutative.
\end{remark}
In the following sections we will handle the general case of a cartesian
product of an arbitrary number, say $n$, of subvarieties. As in the
reformulations of Elekes-Szab\'o's statements expounded above it is easy to
see that a subvariety without power-saving leads to a tuple
$(\a_1,\ldots,\a_n)$ such that each $\a_i$ is $cgp$, belongs to $K^d$ and has
$\bdl(\a_i) \leq d$, such that
$$\bdl(\a_1,\ldots,\a_n)=\dimo(\a_1,\ldots,\a_n).$$
In Sections \ref{sec:proj} and \ref{sec:varieties} we will forget for a moment
the original combinatorial problem and focus entirely on the study of these
tuples. Then in Section \ref{sec:asymptotic} we will return to combinatorics
and give a proof of Theorem \ref{thm:main}.
\iffalse
\subsection{Recovering the Elekes-Szab\'o group existence theorem}
This subsection is not required for the proof of our main results.
In it, we show that Lemma~\ref{lem:cgpCoherentLinearity} already suffices to
recover the Elekes-Szab\'o group existence thereom \cite[Main
Theorem]{ES-groups}, in a slightly stronger form and with an explicit ``gap''.
Apart from the gap, this is a special case of our main theorem (see
Remark~\ref{rmk:mainCodim1}). We present it primarily as a warm-up to the
proof of the main theorem, demonstrating in a more explicit way some of the
ideas in the proof of our main result. In particular, the application of the
(abelian) group configuration theorem is more direct in this case.
The fact that modularity originating from incidence bounds allows one to
deduce versions of the Elekes-Szab\'o theorem in this way was first observed
by Hrushovski \cite[Proposition~5.21]{Hr-psfDims}.
The original proof of Elekes-Szab\'o also went via a version of the group
configuration theorem.
\begin{theorem} \label{thm:coherentES}
Suppose $a_1,a_2,a_3 \in K^{<\infty}$ with $a_1a_2a_3$ coherent,
and $\dimo(a_i) = k$,
and $\dimo(a_1a_2a_3) = 2k = \dimo(a_ia_j)$ for $i\neq j$.
Then there is an abelian algebraic group $G$ over $C_0$,
and $\gamma_1,\gamma_2,\gamma_3 \in G(K)$
such that $\gamma_i$ is generic in $G$ over $C_0$,
and $\gamma_1+\gamma_2+\gamma_3 = 0$,
and $\acl^0(a_i)=\acl^0(\gamma_i)$.
\end{theorem}
\begin{proof}
For notational convenience, define $a := a_1$ and $b := a_2$ and $c := a_3$.
Let $a'b' \equiv _{c} ab$ with $a'b' \dind_{c} ab$;
such $a'$ and $b'$ exist by Remark~\ref{fact:ideal}.
Then by Lemma~\ref{lem:cgpCoherenceAmalg},
$aa'cbb'$ is a coherent tuple,
and $a'b' \ind^0_{c} ab$.
Let $e := \Cb^0(aa'/bb')$ and $e' := \Cb^0(ab'/a'b)$.
Since $a \in \acl^0(a'bb')$,
it follows from Lemma~\ref{lem:cgpCoherentLinearity} that
$\dimo(e) = k = \dimo(e')$.
The remainder of the proof works entirely in the reduct to ACF, and is
similar to \cite[Proof of Proposition 10]{HrUnimod}.
We show that $aa'bb'ee'c$ forms an abelian group configuration.
Consider the following diagram.
\[ \xymatrix{
& & &&&& c \ar@{-}'[ddll][dddlll] \ar@{-}'[dlll][ddllllll] \\
& & & b \ar@{-}'[d][dd] & & & \\
a & & & e' & b' \ar@{-}'[l][llll] & & \\
& & & a' && & \\
&& & & & & e \ar@{-}'[uull][uuulll] \ar@{-}'[ulll][uullllll] } \]
It follows from the definitions of the elements and $\dimo(e)=k=\dimo(e')$
that each point has rank $k$.
We claim that if $f_1f_2f_3$ is a collinear triple,
then $\dimo(f_1f_2f_3) = 2k = \dimo(f_if_j)$ for $i \neq j$.
% then any two points are
% interalgebraic over the third, i.e.\ $f_i \in \acl(f_j,f_k)$ whenever
% $\{i,j,k\}=\{1,2,3\}$.
This is clear for $abc$ and $a'b'c$.
For $aa'e$, note first that $a \ind^0 a'$ and so $\dimo(aa')=2k$.
Meanwhile, $a \ind^0 bb'$ and so $a \ind^0 e$, and hence $\dimo(ae)=2k$;
similarly, $\dimo(a'e)=2k$.
Finally, recalling that $e=\Cb^0(aa'/bb')$, we have
$\dimo(aa'e)=\dimo(aa'/e)+\dimo(e)=\dimo(aa'/bb')+\dimo(e)=k+k=2k$
For $bb'e$, note that $a$ and $c$ are interalgebraic over $b$ and also over
$b'e$ (via $a'$); so $\dimo(ac/bb'e)=\dimo(ac/b)=\dimo(ac)-\dimo(b)$ and
$\dimo(ac/bb'e)=\dimo(ac/b'e)$, so $\acl^0(b) = \acl^0(\Cb^0(ac/bb'e))
\subset \acl^0(b'e)$, so $b \subset \acl^0(b'e)$. Similarly $b' \subset
\acl^0(be)$, and meanwhile $e = \Cb^0(aa'/bb') \subset \acl^0(bb')$. So
$\dimo(be) = \dimo(bb'e) = \dimo(b'e)$,
and $\dimo(bb'e) = \dimo(bb') = 2k$.
By symmetry, the same arguments apply to the remaining lines $ab'e'$ and
$a'be'$.
The rank of the whole diagram is $\dimo(aa'bb'ee'c)=\dimo(abb')=3k$. Any
non-collinear triple other than $\{c,e,e'\}$ has the whole diagram in its
$\acl^0$, hence is independent.
So we conclude by Fact~\ref{fact:abGrpConf}.
\end{proof}
\begin{remark} \label{rmk:ESGap}
Using Lemma~\ref{lem:SzT}(i),
the assumption of coherence in Theorem~\ref{thm:coherentES} can be replaced
by the assumption that each $a_i$ is $cgp$ and $\bdl(a_i)\leq 1$,
and $\bdl(a_1a_2a_3) \geq 2-\eta$ where $\eta := \frac1{8k+6}$.
Indeed, setting again $a := a_1, b := a_2, c := a_3$,
and letting $a'b' \equiv _{c} ab$ with $a'b' \dind_{c} ab$,
we have $\bdl(abcb'a') \geq 2(2-\eta)-1 = 3-2\eta$,
and then since $\eta < \frac12$ we deduce from $cgp$ that
$a'b' \ind^0_{c} ab$.
Now $\bdl(aa'/bb') \geq \bdl(abcb'a') - \bdl(bb') \geq 3-2\eta-2 = 1-2\eta$.
Also $\bdl(aa') \geq \bdl(aa'/c) = \bdl(a/c)+\bdl(a'/c) \geq 2*((2-\eta)-1)
= 2-2\eta$,
and similarly $\bdl(bb') \geq 2-2\eta$.
Let $e := \Cb(aa'/bb')$,
and suppose $f := \dimo(e)>k$.
Then $\dimo(bb'/e) 1$
and $d := 2(f+1) \leq 2(2k+1)$, we have
\begin{align*} \frac 1{2d-1} (d\bdl(aa') + 2(d-1)\bdl(e)) &=
\frac{(1+\beta)d-\beta}{2d-1}\bdl(aa') \\
&= (\frac{(1+\beta)d-1}{2d-1} - \frac {\beta-1}{2d-1})\bdl(aa') \\
&< (\frac{1+\beta}{2} - \frac {1-2\eta}{8k+1})\bdl(aa') \\
&= \frac12\bdl(aa') + \bdl(e) - \frac {1-2\eta}{8k+3})\bdl(aa') \\
&\leq \frac12\bdl(aa') + \bdl(e) - 2\eta ,
\end{align*}
since
\[ \frac {1-2\eta}{8k+3}\bdl(aa') - 2\eta \geq \frac{(1-2\eta)(2-2\eta) -
2(8k+3)\eta}{8k+3} \geq \frac{2(1-(8k+6)\eta)}{8k+3} = 0 .\]
% Note: could slightly improve the estimate by not dropping the quadratic
% terms here.
Meanwhile, since $\eta<\frac14$, we have
$\bdl(e) \geq 2 - 2\eta > 1 + 2\eta \geq \frac12\bdl(aa') + 2\eta$,
so $\bdl(aa') < \frac12\bdl(aa') + \bdl(e) - 2\eta$.
Also $\bdl(e) < \frac12\bdl(aa') + \bdl(e) - 2\eta$,
since $\eta < \frac14 \leq \frac12(1-\eta) \leq \frac14\bdl(aa')$.
But \[ \bdl(aa' e) \geq \bdl(aa'/bb') + \bdl(e) \geq (1-2\eta) + \bdl(e)
\geq \frac12\bdl(aa') + \bdl(e) - 2\eta ,\]
so \[ \bdl(aa'e) \geq \max \left({\frac 1{2d-1} (d\bdl(aa') +
2(d-1)\bdl(e)),
\bdl(aa'), \bdl(e)}\right) .\]
As in the proof of Lemma~\ref{lem:cgpCoherentLinearity},
this contradicts Lemma~\ref{lem:SzT}(i).
So $\dimo(e)=k$, and similarly $\dimo(e')=k$,
and we may now proceed exactly as in the proof of
Theorem~\ref{thm:coherentES} above.
This gap $\eta = \frac1{8k+6}$ is not optimal; for $k=1$, a gap of
$\eta=\frac16$ was obtained by Wang \cite{Wang-gap} and independently by
Raz, Sharir, and de Zeeuw \cite{RSZ-ESR}.
\end{remark}
Applying Lemma~\ref{lem:powersavingCoherent} below, we immediately deduce from
Theorem~\ref{thm:coherentES} the following $n=3$ case of
Theorem~\ref{thm:main}.
\begin{corollary} \label{cor:ES}
Suppose $V \subset W_1\times W_2 \times W_3$ are irreducible complex
varieties, and $\dim(W_i)=k$ and $\dim(\pi_{ij}(V)) = 2k = \dim(V)$ for all
$i\neq j \in \{1,2,3\}$.
Suppose $V$ admits no power-saving. Then $V$ is in co-ordinatewise
correspondence with the graph $\Gamma_+ \subset G^3$ of the group operation
of a commutative algebraic group $G$.
\end{corollary}
\begin{remark} \label{rmk:ESGap2}
By Remark~\ref{rmk:ESGap}, it suffices to assume that $V$ does not admit a
power-saving by any $\epsilon > \eta := \frac1{8k+6}$.
In this form, Corollary~\ref{cor:ES} slightly strengthens \cite[Main
Theorem]{ES-groups} by requiring only coarse general position rather than
general position, by obtaining commutativity of the corresponding algebraic
group, and by giving this explicit value for $\eta$.
\end{remark}
\fi
\end{parag}
\section{Necessity of general position}
\label{gpNecessity}
We give an example showing that Theorem~\ref{cor:ES} fails dramatically if we
weaken too far the coarse general position assumption in the definition of
power-saving. Indeed varieties which are not even in correspondence with a
group operation can then have no power-saving even when the finite sets are
assumed to be say in weak general position, namely assuming $\bdl$ always
drops when there is an algebraic dependence.
% What I'm not saying: if we completely remove the cgp assumption then there
% are silly counterexamples. We at least have to assume some Zariski denseness
% for the conditions on dimension to have any meaning, but even that isn't
% enough because we have silly examples like
% (a,b)*(c,d) = (f(a,c),b+d) where f(a,c) is arbitrary satisfying f(x,x)=x,
% with X_N := \bigcup_{n_F)$ of dimension $\dim(V)$,
and the associated geometry $\P(V) := \P(V, \left<{\cdot}\right>_F)$ is the
projective space of $V$; it also has dimension $\dim(V)$, and it is a
modular geometry.
\end{example}
\begin{example}[Algebraic closure]
An algebraically closed field $K$ equipped with field-theoretic algebraic
closure over the prime field forms a pregeometry $(K,\acl)$. The dimension
is the transcendence degree over the prime field. If $\dim(K) \geq 3$ then
the associated geometry is not modular, as can be seen by considering a
generic solution to $b=c_1a+c_2$, see \cite[Appendix~C.1]{TentZiegler}.
\end{example}
\begin{example}[Algebraic closure on tuples]\label{alg-tuples}
If $C_0 \leq K$ are algebraically closed fields, the set of all tuples
$K^{<\infty}$ equipped with the algebraic closure $\acl^0$ over $C_0$ forms a
closure structure, where the closure of a subset $A \subset K^{<\infty}$ is
$\acl^0(A)^{<\infty}$ as defined in $(\ref{closalg})$. But it is in general
not a pregeometry.
\end{example}
In the sequel we will only consider closure operators of the types described
in the above examples.
%The following is \cite[Lemma~C.1.11]{TentZiegler}.
%Fact{mod2suff}:
% Suppose $(P,\cl)$ is such that \eqref{eq:mod} holds when $\dim(A)=2$.
% Then $(P,\cl)$ is modular.
% .
Let $(P,\cl)$ be a modular geometry.
Then $a,b \in P$ are non-orthogonal
if and only if there exists $c \in P \setminus \{a\}$ such that $a \in
\cl(\{b,c\})$.
In other words, $a=b$ or $|\cl(\{a,b\})|>2$. It is easy to see from modularity
that this is an equivalence relation.
\begin{example}[Abstract projective space] An \defnstyle{abstract projective
space} is a pair $(P,L)$ of sets, where $P$ is the set of points and $L$
the set of lines, a unique line passes through every two distinct points,
every line has at least three points and \emph{the Veblen axiom} holds:
given four distinct points $a,b,c,d$, if the lines $ab$ and $dc$ intersect,
then so do $ad$ and $bc$. Any such abstract projective space gives rise to a modular
geometry on $P$ in which the closure of a subset is the union of all lines passing
through two points in the subset. Conversely any modular geometry in which
every pair of points is non-orthogonal gives rise to an abstract projective
space with the same set of points and with the set of lines being the set of
closures of pairs of distinct points.
\end{example}
%{\bf comment: I am under the impression that we do not use the full force of
%all the definitions above -- of pregeometries, geometries, dimension,
%modularity, etc -- and that we could instead only talk about abstract
%projective spaces as in the previous example; I'm not suggesting we do this
%however, because it's also nice to use this terminology and not very costly.}
%\mbcomment{In the end yes, but I think it's elucidatory to separate the
%argument into two stages as we do: first we see that coherence yields a
%(pre)geometry for simple abstract reasons, and then Szemeredi-Trotter shows
%that it's modular. In particular, the first stage goes through in positive
%characteristic, and there the geometry isn't always modular (and it's natural
%to ask when it is).}
We now recall the classical Veblen-Young co-ordinatisation theorem of
projective geometry, which characterises modular geometries.
\begin{fact} \label{fact:modStructure}
If $(P,\cl)$ is a modular geometry,
and every two points $a,b \in P$ are non-orthogonal,
and $\dim(P) \geq 4$,
then $P$ is isomorphic to a projective space $\P(V)$,
where $V$ is a vector space over a division ring.
More generally,
if $(P,\cl)$ is a modular geometry,
then non-orthogonality is an equivalence relation,
and $(P,\cl)$ is the sum of the subgeometries on its non-orthogonality
classes,
each of which has dimension $\leq 2$, or is a projective space over a
division ring, or is a non-Desarguesian projective plane.
\end{fact}
\begin{proof}
This is a consequence of the classical Veblen-Young co-ordinatisation
theorem for projective geometries. Veblen's axiom is a direct consequence of
modularity.
We refer to \cite[Theorem~3.6]{Cameron-pps} for a statement which directly
implies the stated result and for an overview of the proof,
and to \cite[Chapter~II]{Artin-geometricAlgebra} for a detailed proof of the
co-ordinatisation theorem for Desarguesian projective planes.
\end{proof}
In our applications the geometries will be modular and infinite dimensional.
So by the above they will be sums of projective geometries over division
rings.
\subsection{Coarse general position, coherence and modularity}
We recall the notion of coarse general position for tuples introduced in \S
\ref{cgp-parag}. We keep the notation and setup of Section \ref{sec:setup}.
\begin{definition} \label{defn:cgpType}
A tuple $\c \in K^{<\infty}$ is said to be in \defnstyle{coarse general
position}
(or \defnstyle{is cgp})
if for any $B \subset K^{<\infty}$,
if $\c \nind^0 B$ then $\bdl(\c/B)=0$.
\end{definition}
Recall that $K^{<\infty}$ is the set of all tuples of elements from $K$ and $c
\nind^0 B$ means that $\dimo(\c/B)<\dimo(\c)$, where $\dimo(\c/B)$ denotes as
earlier the transcendence degree of the tuple $\c$ over the field $C_0(B)$
field generated by all co-ordinates of elements from $B$, where $C_0\leq K$ is
the base field defined in \ref{notn:baseField}. The coarse dimension $\bdl$
was defined in $(\ref{defdel})$.
% Eventually we'll want to consider cgp types with varying bases, since any
% type can be analysed in such cgp types. Here we're concentrating on a single
% base.
%\begin{remark}
% This definition is convenient in the non-standard setting, but it does not
% translate cleanly to a finitary setting. It corresponds to a somewhat
% weaker property for finite sets than the earlier definition of $\tau$-cgp.
% We address this issue in Lemma~\ref{lem:cgpComparison}.
%\end{remark}
%(Note: to make this fit with earlier notions of general position, we should
%assume we're working over a $\dcl$-closed set.)
\begin{definition}
A subset $P \subset K^{<\infty}$ is said to be \defnstyle{cgp-coherent} if
every $\a \in P$ is $cgp$ and $\bdl(\a_1,\ldots,\a_n) =
\dimo(\a_1,\ldots,\a_n)$ for all choices of $\a_1,\ldots,\a_n \in P$.
\end{definition}
In this paper, we abbreviate `cgp-coherent' to just `\defnstyle{coherent}'.
% We often leave the coherence constant $\alpha$ implicit.
We will also say that a tuple of tuples from $K^{<\infty}$ is coherent when
the set of its elements is coherent.
% $\a=(a_i)_i \in \Keqztups$ of elements of $\Keqz$ is coherent if the set
% $\{a_i\}_i$ of its elements is coherent.
\begin{remark}The term ``coherent'' is borrowed from
\cite[Section~5]{Hr-psfDims}, where it is used in a parallel context to
refer to the same idea that a pseudo-finite dimension notion is in accord
with transcendence degree.
\end{remark}
%As long as we are working with only one coherence constant, which in fact is
%the case in what we do in this paper, we can rescale and assume $\alpha=1$.
%But we will not do this.
% Maybe we should?
% (And then we did)
%Remark:
% Any $\bdl$-independent sequence of realisations of a cgp type is coherent.
% .
We are now ready to state the main result of this section.
\begin{theorem} \label{cor:coherentGeom}
Suppose $P \subset K^{<\infty}$ is coherent. Then
$(P;\acl^0\negmedspace\restriction_{P})$ is a pregeometry. Moreover $P$
extends to a coherent $P' \subset K^{<\infty}$ such that the geometry
$\G_{P'} := \P(P';\acl^0\negmedspace\restriction_{P'})$ is a sum of
$1$-dimensional geometries and infinite dimensional projective geometries
over division rings.
\end{theorem}
Here the closure operator is simply the restriction to $P'$ of the algebraic
closure $\acl^0$ as in Example \ref{alg-tuples}, namely if $A \subset P'$,
$\acl^0\negmedspace\restriction_{P'}(A)$ is the set of tuples in $P'$ whose
co-ordinates are algebraic over the subfield of $K$ generated by $C_0$ and the
set of all co-ordinates of all tuples from $A$.
The proof of Theorem \ref{cor:coherentGeom} will proceed in two steps. First
we will show that if $P \subset K^{<\infty}$ is coherent, then its coherent
algebraic closure $\ccl(P): = \{ \x \in \acl^0(P)^{<\infty} : \x \textrm{ is
cgp and } \bdl(\x) = \dimo(\x) \}$ is also coherent. And second we will
prove that if $P=\ccl(P)$ is coherent, then
$(P;\acl^0\negmedspace\restriction_P)$ is a modular pregeometry. The latter
step will use the incidence bounds \`a la Szemer\'edi-Trotter recalled in
Section \ref{subsec:incidence}. Theorem \ref{cor:coherentGeom} will then
follow by applying the Veblen-Young theorem recalled in Fact
\ref{fact:modStructure} above to the projectivisation of
$(P;\acl^0\negmedspace\restriction_P)$.
The rest of this section is devoted to the proof of Theorem
\ref{cor:coherentGeom}.
\begin{parag}{\bf Properties of coherent sets.}
\begin{proposition}\label{pregeo}
If $P \subset K^{\infty}$ is coherent then
$(P;\acl^0\negmedspace\restriction_P)$ is a pregeometry.
\end{proposition}
\begin{proof}
We verify exchange, the other properties being immediate.
Suppose $\b \in \acl^0(A \cup \{\c\}) \setminus \acl^0(A)$ for $A \subset P$
and $\b,\c \in P$. So $\b \nind^0_{A} \c$. By symmetry of $\ind^0$ (see \S
\ref{ind}), we get $\c \nind^0_{A} \b$. Now the next lemma forces $\c \in
\acl^0(A \cup \{\b\})$.
\end{proof}
\begin{lemma}\label{lem:cohFullEmb} Suppose $P \subset K^{<\infty}$ is
coherent. Let $\c \in P$ and $A,B \subset P$. Then either $\c \ind^0_A B$,
or $\c \in \acl^0(A \cup B)$.
\end{lemma}
\begin{proof}
Suppose $\c \nind^0_A B$.
Then in particular $\c \nind^0 (B \cup A)$,
and so $\c \nind^0 \b\a$ for some tuples $\b \in B^{<\infty}$ and $\a \in
A^{<\infty}$.
Since $\c$ is $cgp$ this implies $\bdl(\c/\b\a)=0$. Since $P$ is coherent,
$\dimo(\c\b\a)=\bdl(\c\b\a)$ and $\dimo(\b\a)=\bdl(\b\a)$ so by additivity
of $\dimo$ and $\bdl$ we get $\dimo(\c/\b\a)=0$. Hence $\c \in \acl^0(A
\cup B)$.
\end{proof}
The next lemma will be used to form coherent sets.
\begin{lemma}\label{lem1} Let $\a_1,\ldots,\a_n \in K^{<\infty}$. Assume that
each $\a_i$ is $cgp$ and $\bdl(\a_i) \leq \dimo(\a_i)$. Then for every $C
\subset K^{<\infty}$ we have:
\begin{equation}\label{upb}\bdl(\a_1,\ldots,\a_n/C) \leq
\dimo(\a_1,\ldots,\a_n/C).
\end{equation}
Moreover $\bdl(\a_1,\ldots,\a_n)= \dimo(\a_1,\ldots,\a_n)$ if and only if
$\{\a_1,\ldots,\a_n\}$ is coherent.
\end{lemma}
\begin{proof} The proof is by induction on $n$. Suppose first $n=1$. We have a
$cgp$ $\a_1 \in K^{<\infty}$ such that $\bdl(\a_1) \leq \dimo(\a_1)$ and we
need to show that $\bdl(\a_1/C) \leq \dimo(\a_1/C).$ If $\a_1 \ind^0 C$,
then $\dimo(\a_1/C)=\dimo(\a_1)$, so the desired inequality follows
immediately. On the other hand, if $\a_1 \nind^0 C$, then by $cgp$
$\bdl(\a_1/C)=0$, so the desired inequality is then obvious.
Suppose $(\ref{upb})$ holds for $n-1$ tuples and any $C$. Let $\x=\a_1 \ldots
\a_{n-1}$. $$\bdl(\x\a_n/C) = \bdl(\x/C \cup\{\a_n\}) + \bdl(\a_n/C) \leq
\dimo(\x/C \cup\{\a_n\}) + \dimo(\a_n/C)= \dimo(\x\a_n/C),$$
where we applied the induction hypothesis and the case $n=1$.
Finally we turn to the last claim of the lemma. Suppose
$\bdl(\a_1\ldots\a_n)=\dimo(\a_1\ldots\a_n)$. We need to show that
$\bdl(\x)=\dimo(\x)$ for all concatenated tuples $\x$ made of subtuples from
$\{\a_1,\ldots,\a_n\}$. Note that for every tuple of $\a_i$'s the quantities
$\bdl$ and $\dimo$ depend only on the subset of $\a_i$'s appearing in the
tuple (see Fact \ref{set-dep}), so up to relabelling co-ordinates we may
assume that $\x=\a_1\ldots \a_i$ for $i\in [1,n]$. Let
$\y:=\a_{i+1}\ldots\a_n$. Then by assumption $\bdl(\x\y)=\dimo(\x\y)$. By
$(\ref{upb})$ we have $\bdl(\y/\x) \leq \dimo(\y/\x)$ and $\bdl(\x) \leq
\dimo(\x)$. Hence by additivity we conclude that the last two inequalities are
equalities. This ends the proof.
\end{proof}
Finally we record one last observation, which will be useful in the next
paragraph.
\begin{lemma}\label{lem2} If $P \subset K^{<\infty}$ is coherent and $\x \in
\acl^0(P)^{<\infty}$. Then $\bdl(\x) \geq \dimo(\x)$.
\end{lemma}
\begin{proof} Pick $\a_1,\ldots,\a_n \in P$ such that $\x \in
\acl^0(\{\a_1,\ldots,\a_n\})^{<\infty}$ and concatenate the $\a_i$'s in
$\a:=\a_1\ldots\a_n$. Then $\dimo(\a)=\dimo(\a\x)$. By additivity
$\dimo(\a\x)=\dimo(\x)+\dimo(\a/\x)$ and $\bdl(\a\x)=\bdl(\x)+\bdl(\a/\x)$.
By coherence of $P$ we have $\bdl(\a)=\dimo(\a)$. But $\bdl(\a/\x) \leq
\dimo(\a/\x)$ by Lemma \ref{lem1}. So $$\bdl(\x) \ge \bdl(\a\x) -
\dimo(\a/\x) \ge \bdl(\a) - \dimo(\a/\x) = \dimo(\a) - \dimo(\a/\x) =
\dimo(\x).$$
\end{proof}
\end{parag}
\begin{parag}{\bf The Veblen axiom and incidence bounds.} In this paragraph we
exploit the Szemer\'edi-Trotter-type bounds described in Subsection
\ref{subsec:incidence} in order to show that the pregeometry
$(P;\acl^0\negmedspace\restriction_P)$ satisfies the Veblen axiom of
projective geometry.
\begin{proposition}\label{lem:cgpCoherentLinearity} Assume $P\subset
K^{<\infty}$ is coherent, let $\a_1,\a_2 \in P$ and $B \subset P$. Assume
that $\a_1,\a_2 \notin \acl^0(B)^{<\infty}$, but $\a_2 \in\acl^0(\{\a_1\}
\cup B)^{<\infty}\setminus \acl^0(\a_1)^{<\infty}$. Then there is $\d \in
\acl^0(B)^{<\infty}$ such that $\d \in \acl^0(\a_1,\a_2)^{<\infty}$ and
$\bdl(\d)=\dimo(\d)=\dimo(\a_1)=\dimo(\a_2)$.
\end{proposition}
\begin{proof} For brevity set $\a:=\a_1\a_2$. Without loss of generality we
may assume $B$ is finite and $B=\{\a_3,\ldots,\a_n\}$. We set
$\b=(\a_3,\ldots,\a_n)$. First we check that $\a_1 \in
\acl^0(\a_2,\b)^{<\infty}$, $\a_i \ind^0 \b$ and $\a_1 \ind^0 \a_2$, and
that if $k:=\dimo(\a_1)$, then $\dimo(\a_2)=k$ and $\dim(\a)=2k$. The first
property follows from the exchange property of pregeometries and from
Proposition \ref{pregeo}. Lemma \ref{lem:cohFullEmb} tells us that $\a_i
\ind^0 \b$, since $\a_i \notin \acl^0(\b)^{<\infty}$. For the same reason
$\a_1 \ind^0 \a_2$. Then $\dimo(\a\b)=\dimo(\a_1\b)=\dimo(\a_2\b)$ is equal
to both $\dimo(\a_1)+\dimo(\b)$ and $\dimo(\a_2)+\dimo(\b)$. Hence
$\dimo(\a_2)=k$ and $\dimo(\a)=2k$. This also shows that $\dimo(\a/\b)=k$.
Let $\d \in K^{<\infty}$ be a canonical base for $\a$ over $\b$ (see \S
\ref{parag:canonical}). By definition $\d \in \acl^0(\b)^{< \infty}$. We will
show that this $\d$ satisfies the desired conclusion. By definition we have
$\dimo(\a/\d)=\dimo(\a/\b)=k$. Hence $\dimo(\d)=\dimo(\a\d)-\dimo(\a/\d)\ge
\dimo(\a)-k=k$. By Lemma \ref{lem2} $\bdl(\d) \geq \dimo(\d)$. Hence we are
left to show the upper bound $\bdl(\d) \leq k$.
To this end let $V$ be the locus of the tuple $\a\d$, i.e.\ $V=\loc^0(\a,\d)$,
let $X_1\subset K^{<\infty}$ be the type of $\a$, i.e.\ $X_1=\tp(\a)(K)$, let
$X_2=\tp(\d)(K)$ and finally let $X=(X_1 \times X_2) \cap V$. We wish to apply
the Szemer\'edi-Trotter bound of Lemma \ref{lem:SzT} to this data.
For this we first show that $\bdl(X(\d_1)\cap X(\d_2))=0$ for all $\d_1, \d_2
\in X_2$ with $\d_1 \neq \d_2$, so that $\beta=0$ in this lemma. Recall that
$X(\d_1):=\{\a' \in X_1 : \a'\d_1 \in V\}$. By Fact \ref{fact:ideal} we may
find $\a'=\a'_1\a'_2 \in X(\d_1)\cap X(\d_2)$ such that
$\bdl(\a'/\d_1\d_2)=\bdl(X(\d_1)\cap X(\d_2))$. Then $\a'\d_i \in V$ for
$i=1,2$. We are thus in the setting of Lemma \ref{lem:cb}. Since $\d_1 \neq
\d_2$ we conclude that $\dimo(\a'/\d_1\d_2)k$. Then $\dimo(\b/\d)=\dimo(\b)
-\dimo(\d)<(n-3)k$. Writing $$\dimo(\b/\d) = \sum_{i=3}^n \dimo(\a_i /
\a_3,\ldots,\a_{i-1} \d),$$
a sum of $n-2$ terms, each $\leq k$, we see that at least two terms must be
$ \frac{1}{4(n-2)},$$
a contradiction to our assumption. This contradiction shows that $\dimo(\d)=k$
and ends the proof.
\end{proof}
\fi
\end{parag}
\begin{parag}{\bf Proof of Theorem \ref{cor:coherentGeom}.}
Here we show Theorem \ref{cor:coherentGeom}.
Lemma~\ref{lem:cgpCoherentLinearity} will help us find a modular geometry
explaining algebraic dependence on a coherent set. This is the engine behind
our main results. The idea comes from \cite[Subsection~5.17]{Hr-psfDims}, and
the context is essentially that of \cite[Remark~5.26]{Hr-psfDims}.
\begin{definition}
For $P \subset K^{<\infty}$, define the \defnstyle{coherent algebraic
closure} by
\[ \ccl(P) := \{ \a \in \acl^0(P)^{<\infty} : \a \textrm{ is cgp and }
\bdl(\a) = \dimo(\a) \} .\]
\end{definition}
\begin{lemma} \label{lem:cgpCoherentClosure}
If $P$ is coherent, then so is $\ccl(P)$.
\end{lemma}
\begin{proof}
We need to show that $\bdl(\a_1\ldots\a_n)=\dimo(\a_1\ldots\a_n)$ for any
$\a_i$'s from $\ccl(P)$. We proceed by induction on $n$. This holds when
$n=1$ by the definition of $\ccl(P)$. Set $\x:=\a_1\ldots\a_{n-1}$. By
induction hypothesis and Lemma \ref{lem1} $\{\a_1,\ldots,\a_{n-1}\}$ is
coherent and $\bdl(\x/\a_n) \leq \dimo(\x/\a_n)$. So by additivity
$$\bdl(\x\a_n) =\bdl(\x/\a_n)+\bdl(\a_n) \leq
\dimo(\x/\a_n)+\dimo(\a_n)=\dimo(\x\a_n).$$ But Lemma \ref{lem2} implies
that $\bdl(\x\a_n) \geq \dimo(\x\a_n)$. This ends the proof.
\end{proof}
Clearly $\ccl(\ccl(P)) = \ccl(P)$.
We say $P \subset K^{<\infty}$ is \defnstyle{coherently algebraically closed}
if $P$
is coherent and $P=\ccl(P)$.
\begin{proposition} \label{thm:cgpCoherentModularity}
Suppose $P \subset K^{<\infty}$ is coherently algebraically closed.
Then the pregeometry $(P;\acl^0\negmedspace\restriction_P)$ is modular.
\end{proposition}
\begin{proof}
We must show that if $B \subset P$ and $\a_1,\a_2 \in P\setminus
\acl^0(B)^{<\infty}$ are such that $\a_1 \in \acl^0(B \cup
\{\a_2\})^{<\infty}$, then $\a_1 \in \acl^0(\{\d,\a_2\})^{<\infty}$ for some
$\d \in P \cap \acl^0(B)^{<\infty}$. We may assume without loss of
generality that $B$ is finite, say $B=\{\a_3,\ldots,\a_n\}$. This is the
situation of Proposition \ref{lem:cgpCoherentLinearity} from which we
conclude that there is an integer $k$ such that
$k=\dimo(\a_1\a_2/\d)=\dimo(\a_2)=\dimo(\a_1)=\dimo(\d)$ for some $\d \in
\acl^0(\{\a_1,\a_2\})^{<\infty}$.
We are left to show that $\d \in P$, and since we already know that
$\bdl(\d)=\dimo(\d)$ and $\d \in \acl^0(P)$, we are only left to check that
$\d$ is $cgp$.
To this end assume that $\d$ is not $\acl^0$ independent from $E$, for some $E
\subset K^{<\infty}$. We need to show that $\bdl(\d/E)=0$. By Fact
\ref{fact:ideal} we may pick $\a_1',\a_2' \in K^{< \infty}$ such that
$\tp(\a_1',\a_2'/\d) = \tp(\a_1,\a_2/\d)$ and $\bdl(\a_1',\a_2'/E\d) =
\bdl(\a_1',\a_2'/\d)$. For brevity write $\a:=\a_1\a_2$ and $\a'=\a_1'\a_2'$.
By additivity of $\bdl$ we may write:
$$\bdl(\d/E)=\bdl(\d/\a'E)+\bdl(\a'/E) - \bdl(\a'/\d E).$$
Let us examine the three terms on the right hand side.
The first term $\bdl(\d/\a'E)$ is zero since $\d \in \acl^0(\a')^{<\infty}$,
because $\d \in \acl^0(\a)^{<\infty}$.
The second term is equal to $\bdl(\a_1'/\a_2' E) + \bdl(\a_2'/ E)$. We claim
that this is at most $k$. Note that $\a_1',\a_2'$ are $cgp$ because
$\a_1',\a_2'$ are $cgp$ (see Remark \ref{cgp-invariance}). By $(\ref{upb})$ it
is enough to show that one of these terms is zero. Hence by the $cgp$ we only
need to check that either $\a_1' \nind^0 \a_2' E$ or $\a_2' \nind^0 E$. In
other words that $\a_1'\a_2' \nind^0 E$. This is indeed the case for otherwise
$\d$ would be independent from $E$, because $\d \in \acl^0(\a')^{<\infty}$.
Finally let us turn to the third term. Since $\d \in \acl^0(B)^{<\infty}$ by
Fact \ref{alg-ineq} we have $$ \bdl(\a'/\d) = \bdl(\a/\d) \geq
\bdl(\a/\acl^0(B)) = \bdl(\a/B).$$ On the other hand since $\a_1,\a_2,B$ lie
in $P$ and $P$ is coherent we have $\bdl(\a/B)= \dimo(\a/B)$. By Proposition
\ref{lem:cgpCoherentLinearity} this is $k$. Hence $\bdl(\a'/\d E)
=\bdl(\a'/\d) \geq k$. This concludes the proof.
% ^Note: it's for this argument that we have to use cgp rather than gp, and
% so need the stronger form of Szemer\'edi-Trotter.
\end{proof}
\begin{proof}[Proof of Theorem \ref{cor:coherentGeom}] By Proposition
\ref{pregeo} $(P;\acl^0\negmedspace\restriction_{P})$ is a pregeometry. By
Lemma \ref{lem:cgpCoherentClosure}, enlarging $P$ to $\ccl(P)$ if necessary,
we may assume that $P$ is coherently algebraically closed. By
Proposition~\ref{thm:cgpCoherentModularity} the associated geometry $\G_P$
is modular. Hence the non-orthogonality relation is an equivalence relation.
Up to enlarging $P$ further if necessary, we can assume that each
non-orthogonality class in $\G_P$ of dimension $>1$ has infinite dimension.
This follows from the Lemma \ref{dim-increase} below applied iteratively
countably many times. In each finite dimensional non-orthogonality class we
pick a point $a$ and increase its dimension without altering the other
classes, until all classes are infinite dimensional. Now we conclude by the
Veblen-Young Theorem as recalled in Fact~\ref{fact:modStructure}.
\end{proof}
\begin{lemma}[dimension increase]\label{dim-increase} If $P$ is coherently
algebraically closed and $a,b,c \in P$ are distinct in $\G_P$ and collinear
in the sense that $c \in \acl^0({a,b})$, then there is $a' \in K^{<\infty}$
non-orthogonal to $a$ such that $P':=\ccl(P \cup \{a'\})$ is coherent, $a'
\ind^0 P$, and every $x \in P'$ is either in $P$ or non-orthogonal to $a$.
\end{lemma}
\begin{proof} Note first if $a,b \in P$ are non-orthogonal, then
$\dimo(a)=\dimo(b)$. This is part of the conclusion of Proposition
\ref{lem:cgpCoherentLinearity}, or also follows easily from Lemma
\ref{lem:cohFullEmb}.
Now by Fact \ref{fact:ideal} we can pick $a',c' \in K^{<\infty}$ with
$\tp(a'c'/b)=\tp(ac/b)$ and $\bdl(a'c'/P)=\bdl(ac/b)$. Since $P$ is coherent
$\bdl(a/b)=\dimo(a/b)=\dimo(a)=\bdl(a)$, while since $a,a'$ have the same type
$\bdl(a)=\bdl(a')$ coincides with $\dimo(a')=\dimo(a)$. Also $a'$ and $c'$ are
$cgp$ (see Remark \ref{cgp-invariance}). Since $\tp(a'c'/b)=\tp(ac/b)$ we have
$c' \in \acl^0(P \cup \{a'\})$ and hence $\bdl(c'/Pa')=0$. Similarly,
$\bdl(c/ab)=0$. By additivity we conclude
$\bdl(a'/P)=\bdl(a'c'/P)=\bdl(ac/b)=\bdl(a/b)=\dimo(a)$. Hence also $a' \ind^0
P$ since $a'$ is $cgp$. We conclude $\dimo(a'/P)=\bdl(a'/P)$. It follows from
Lemma~\ref{lem1} and additivity that $P \cup \{a'\}$ is also coherent.
By Lemma \ref{lem:cgpCoherentClosure} $P'=\ccl(P \cup \{a'\})$ is also
coherent. Moreover $b,a',c'$ are collinear so $a'$ is non-orthogonal to $a, b$
and $c$. Finally if $x \in P'$, then $x \in \acl^0(P \cup \{a'\})$, so if $x
\notin P \cup \acl^0(a')$ by modularity there is $y \in P$ such that $x,a',y$
are collinear, hence $a'$ and $x$ are non-orthogonal.
\end{proof}
%Question:
% Can we prove coherent modularity for cgp types of $\dim^0>1$ without
% appealing to Szemer\'edi-Trotter and in arbitrary characteristic?
% (This would make sense from the point of view of analogies with other
% trichotomous model theoretic situations: if modularity fails, we expect to
% obtain a field definably isomorphic to a pseudo-finite subfield, which even
% in positive characteristic should only be possible in dimension 1.)
% .
\begin{remark}
The results of this section go through with the same proofs when $ACF_0$
is replaced by an arbitrary finite U-rank theory in which
Lemma~\ref{lem:SzT} holds.
% Can we generalise that Lemma to other contexts, such as D-varieties, or
% compact complex manifolds, or finite Morley rank groups? When we have
% trichotomy / CBP, as in D-varieties and CCM, can we deduce it directly
% from the ACF case? Or for CCM, can we deduce it from the recent o-minimal
% versions applied to $\R_{\operatorname{an}}$.
\end{remark}
\end{parag}
%Definition:
% Say $P \subset \Keqz$ is \defnstyle{1-based}$^0$
% if for any tuples $\a$ and $\b$ from $P$,
% we have $\acleqz(\Cb^0(\a/\b)) = \acleqz(\a)\cap\acleqz(\b)$.
% .
%Corollary{cor:coherent1basedness}:
% If $P \subset \Keqz$ is coherent then $P$ is 1-based$^0$.
% .
%Proof:
% By Lemma~\ref{thm:cgpCoherentModularity},
% we have $\a \ind^0_{\acleqz\negmedspace\restriction_{\ccl(P)}(\a) \cap
% \acleqz\negmedspace\restriction_{\ccl(P)}(\b)} \b$,
% and so in particular $\a \ind^0_{\acleqz(\a)\cap\acleqz(\b)} \b$.
% So $\Cb^0(\a/\b) \subset \acleqz(\a)\cap\acleqz(\b)$,
% and clearly $\acleqz(\a)\cap\acleqz(\b) \subset \acleqz(\Cb^0(\a/\b))$.
% .
%# Relative notions
%Notation:
% We have worked ``absolutely'' in this section, but we will also need the
% corresponding relative notions. We obtain these by base-change (as defined
% in Notation~\ref{notn:baseField}).
%
% We say $P$ is \defnstyle{coherent over $C$} if $P$ is coherent after
% base-change to $C$, and similarly for \defnstyle{1-based$^0$ over} and
% \defnstyle{cgp over}.
% .
%
%Lemma{coherentBaseChange}:
% **(i) If $a \in \Keqz$ is cgp and $C \subset \Keqz$, then $a$ is cgp over
% $C$.
% **(ii) If $P \subset \Keqz$ is coherent and $Y \subset P$, then $P$ is
% coherent over $Y$.
% *ee*
% .
%Proof:
% **(i)
% If $a \nind^0 C$, then $\bdl(a/C)=0$ by cgp, so certainly $a$ is cgp
% over $C$.
% Else, if $a \nind^0_C D$ then $a \nind^0 CD$ so $\bdl(a/CD)=0$, so
% again $a$ is cgp over $C$.
% **(ii)
% Each $a \in P$ is cgp over $Y$ by (i), and if $\a$ is a tuple from $P$
% then
% $\inc(\a/Y) = 0$ by coherence of $P$. So $P$ is coherent over $Y$ by
% Lemma~\ref{lem:cgpCoherenceExtremal}.
% *ee*
% .
%
%Remark:
% Coherence is not transitive - if $A$ is coherent over $B$ and $B$ is
% coherent, it does not follow that $AB$ is coherent. To see this, consider
% the case of $A = \acleqz(B)$.
% .
%
%We also single out the following useful analogue of
%Lemma~\ref{lem:cgpCoherenceAmalg}.
%Lemma{lem:indCohRel}:
% Suppose $\a \in \Keqztups$ is coherent and $C \subset \Keqz$,
% and $\a \dind C$.
% Then $\a$ is coherent over $C$ (with the same coherence constant),
% and $\a \ind^0 C$.
% .
%Proof:
% Let $\alpha$ be the coherence constant.
% We may assume that no $a_i \in \acleqz(\emptyset )$.
% By additivity and monotonicity of $\bdl$, we have $\bdl(\a'/C)=\bdl(\a')$
% for every subtuple $\a'$ of $\a$.
% In particular $\bdl(a_i/C)=\bdl(a_i)=\alpha\dim^0(a_i)>0$, and so by cgp,
% $a_i \ind^0 C$. So $a_i$ is coherent over $C$.
% Meanwhile $\inc(\a/C) \geq \inc(\a) = 0$.
% So by Lemma~\ref{lem:cgpCoherenceExtremal},
% $\a$ is coherent over $C$,
% and $\inc(\a/C) = 0 = \inc(\a)$ so $\dim^0(\a/C) = \dim^0(\a)$.
% .
\section{Varieties with coherent generics}\label{sec:varieties}
We now show in Proposition \ref{prop:coherentSpecial} below that the locus of
a coherent tuple is a special variety.
This will follow from Theorem~\ref{cor:coherentGeom} and a characterisation of
the projective geometries which can arise from $\acl^0$. We shall give such a
characterisation in Appendix~\ref{appx:EHeq}, generalising a result of
Evans-Hrushovski.
\begin{proposition} \label{prop:coherentSpecial}
Suppose $a_1,\ldots ,a_n \in K^{<\infty}$ are such that $\a=(a_1,\ldots
,a_n)$ is coherent.
Then $\loc^0(\a)$ is a special subvariety of $\prod_i \loc^0(a_i)$.
\end{proposition}
\begin{remark}
Technically, we defined ``special'' only for complex varieties, but
$\loc^0(\a)$ is a variety over $C_0$ and $C_0$ need not come with an
embedding into $\C$. In our main applications in
Section~\ref{sec:asymptotic} below, $C_0$ will come with such an embedding;
more generally, we may take an arbitrary such embedding, or just define
``special'' for varieties over an algebraically closed field $C_0$ by exact
analogy to the definition for varieties over $\C$ in the introduction.
\end{remark}
% \mbcomment{I've avoided using tuple notation for the $a_i$ here, i.e.\ not
% writing $\a=(\a_1,\ldots ,\a_n)$ etc., even though this is what is now done
% in the previous sections. It's really irrelevant to the argument that they
% are tuples; moreover it would be quite distracting to write the $h_i \in
% G(K)$ as tuples, and doing one but not the other would look bizarre.}
\begin{proof}
In this proof we make use of some of the definitions from
Appendix~\ref{appx:EHeq}, applied to the pair of algebraically closed fields
$C_0 \leq K$. In particular, given $x \in K^{<\infty} \setminus
\acl^0(\emptyset )$ we set $\widetilde x := \acl^0(x)$, and we let $\G_K $
be the projectivisation of the closure structure $(K^{<\infty},\acl^0)$
defined in Example \ref{alg-tuples} above, namely
$\G_K := \P(K^{<\infty};\acl^0) = \{ \widetilde x : x \in K^{<\infty}
\setminus \acl^0(\emptyset ) \}$.
We may assume no $a_i$ lies in $\acl^0(\emptyset )$. Indeed, if say $a_1 \in
\acl^0(\emptyset )$, then $\loc^0(\a) = \{ a_1 \} \times \loc^0(a_2,\ldots
,a_n)$, and $\{ a_1 \}$ is special (with the trivial group, which is a
special subgroup of itself), and so it suffices to show that
$\loc^0(a_2,\ldots ,a_n)$ is special.
By Theorem~\ref{cor:coherentGeom}, $\{a_1,\ldots,a_n\}$ extends to a
coherent set $P$ such that $\G_P = \{ \widetilde p : p \in P \} \subset
\G_K$ splits as a sum of 1-dimensional and infinite dimensional projective
geometries over division rings. This induces a corresponding splitting of
$\a$ into subtuples of $a_i$'s, and the locus of $\a$ is the product of
their loci. So it suffices to show that each such locus is special. So we
may assume $\widetilde a_1,\ldots ,\widetilde a_n$ are all contained in a
single summand.
We conclude by showing that $\loc^0(\a)$ is in co-ordinatewise
correspondence with a special subgroup, by finding a commutative algebraic
group $G$ over $C_0$ and generics $h_i \in G(K)$ with $\widetilde a_i =
\widetilde {h}_i$, such that $\loc^0(\h)$ is a special subgroup of $G^n$. By
Remark~\ref{rmk:corrGeneric}, this will suffice.
If $\widetilde {a_i} = \widetilde {a_j}$ for $i \neq j$, we can take $h_j :=
h_i$.
So assume there are no such interalgebraicities. Let $\G_{\a} := \{
\widetilde a_1, \ldots, \widetilde a_n \} \subset \G_P \subset \G_K$.
If $\dim(\G_{\a})=1$ (the ``trivial'' case), then $\a=a_1$ and we may take
$G := \mathbb{G}_a^{\dimo(a_1)}$ and a point $h_1 \in G(K) = K^{\dimo(a_1)}$
with $\widetilde {a_1} = \widetilde {h_1}$.
Else, $\G_{\a}$ embeds in a projective geometry over a division ring,
where moreover by Lemma~\ref{lem:cohFullEmb} the latter geometry is fully
embedded in $\G_K$ in the sense of Definition~\ref{defn:fullyEmbedded}.
So by Proposition~\ref{prop:EHeq},
there is an abelian algebraic group $G$ over $C_0$ with $\dim(G) =
\dimo(a_i)$,
and a division subring $F$ of $\End^0_{C_0}(G)$,
and $\h = (h_1,\ldots ,h_n) \in G(K)^n$ with $\widetilde {h_i} = \widetilde
{a_i}$,
such that (in particular) $\dim_F(\left<{\h/G(C_0)}\right>_F) =
\dim(\G_{\a})$.
Hence $A\cdot(\h/G(C_0))=0$ for some $A \in \Mat_n(F)$ of rank $n -
\dim(\G_{\a})$.
By clearing denominators, we may assume that $A$ has entries from
$\End_{C_0}(G) \cap F$.
Let $\c := A\cdot\h \in G(C_0)^n$. Since $C_0$ is algebraically closed,
$A\cdot\x = \c$ has a solution $\h_0 \in G(C_0)^n$. Replacing $\h$ by
$\h-\h_0$, which does not affect $\widetilde {h_i}$, we may assume $\h \in
\ker(A)$.
Write $\ker(A)^o$ for the connected component of $\ker(A)$.
By further replacing $\h$ by $e\cdot\h$ where $e \in \N$ is the exponent of
the finite group $\ker(A)/\ker(A)^o$, we may assume $\h \in \ker(A)^o$.
Now it is not hard to see, e.g.\ by considering Gaussian elimination,
that $\dim(\ker(A)) = \dim(G)(n-\operatorname{rank}(A))$.
So $\dim(\ker(A)^o) = \dim(G)(n-\operatorname{rank}(A)) = \dimo(\a) =
\dimo(\h)$. Therefore $\loc^0(\h) = \ker(A)^o$ is a special subgroup of
$G^n$ as required.
\end{proof}
\section{Asymptotic consequences}\label{sec:asymptotic}
In this section we first unpeel the ultraproduct construction to show how
coherent tuples correspond to varieties without powersaving. Then, combining
this with Proposition~\ref{prop:coherentSpecial} above and some further
argument, we prove the combinatorial theorems stated in the introduction.
Let $W_i$, $i=1,\ldots ,n$, be irreducible complex algebraic varieties each of
dimension $d$, and let $V \subset \prod_i W_i$ be an irreducible complex
closed subvariety. We first recall the following simple observation, already
mentioned in the introduction.
%Recall Convention~\ref{convn:vars} on how we consider $K$-points of varieties
%as elements of $\Keqz$.
\begin{lemma}[the trivial bound] \label{lem:power-saving}
Let $\tau>d$. There is $C \in \N$ depending only on $\tau$ and $V$ such that
if $X_i \subset W_i$ is in $(C,\tau)$-coarse general position in $W_i$ (see
Def. \ref{defn:taucgp}) and if $|X_i|\leq N^d$,
then $|V \cap \prod_{i\leq n} X_i| \leq O_V(N^{\dim(V)})$.
Furthermore, if $V$ admits no power-saving then $\dim(V)$ is an integral
multiple of $d$.
\end{lemma}
\begin{proof}
We prove this by induction on $n$ and $\dim(V)$, with $C$ and the
multiplicative constant in $O_V$ depending only on the complexity of $V$ and
the $W_i$'s.
For $n=1$ it is clear.
For $n>1$, consider the projection $\pi : V \rightarrow \prod_{id$ the desired bound holds,
and moreover that $V$ admits a power-saving.
Finally if the projection of $V$ to $W_i$ for some $i$ is not dominant, then
has no power saving, then so does $Y$ and $V$ has dominant projections on all
$W_i$'s with $i N_s^{\dim
V-\epsilon}$
and $|X_{i,s} \cap W'_i| \le |X_{i,s}|^\epsilon$ for any $W'_i$ a proper
closed subvariety of $W_i$ of complexity $\leq \tau$.
After enlarging $\L$ if necessary, we may assume that $\prod_{s \rightarrow
\U} X_{i,s} =: X_i$ are definable without parameters in $\L$. Set the scaling
constant $\xi := \lim_{s \rightarrow \U} N_s$.
Then by the above estimates and Lemma~\ref{lem:power-saving},
$\bdl(V \cap \prod_i X_i) = \dim V$.
So by Fact~\ref{fact:ideal}, say $\a=(\a_1,\ldots,\a_n) \in V \cap
\prod_{i=1}^n X_i$ with $\bdl(\a)=\dim V$. By construction each $X_i$ is
$cgp$ in $W_i$, so $\a_i$ is $dcgp$ and hence $cgp$ (Lemma \ref{cgplem}). In
particular $\bdl(\a_i)\leq \dimo(\a_i)$, for $\bdl(\a_i) \leq d$ since $\a_i
\in X_i$ and either $\dimo(\a_i)=d$ or $\a_i$ is contained in a proper
subvariety of $W_i$ defined over $C_0$, which forces $\bdl(\a_i)=0$, since
$X_i$ is $cgp$ in $W_i$. Also $\a$ is generic in $V$, i.e.\ $\dimo(\a)=\dim
V$, for otherwise $\a \in V'$ for some proper irreducible subvariety $V'$ of
$V$ defined over $C_0$ and hence by the trivial bound of
Lemma~\ref{lem:power-saving} we would have $\bdl(\a)\leq \bdl(V' \cap
\prod_{i=1}^n X_i) \leq \dim V -1$. It then follows from Lemma \ref{lem1}
that $\a$ is coherent.
Suppose conversely that, for some $K_s$ and $\xi \in \stR$, we have a tuple
$\a \in V(K)$ which is coherent generic and for each $i$ we have $\a_i \in
X_i$, an $\L$-definable without parameters and $cgp$ subset of $W_i(K)$. To
say that $\a$ is coherent means that $\{\a_1,\ldots,\a_n\}$ is a coherent
set. In particular $\a_i$ is $cgp$ and $\bdl(\a_i)=\dimo(\a_i)$. Since $\a$
is generic in $V$, its projection $\a_i$ is generic in the co-ordinate
projection $\pi_i(V) \subset W_i$. We may assume that this projection is
dominant, for otherwise by $cgp$ we would have $\bdl(\a_i)=0$ and hence
$\dimo(\a_i)=0$, which would mean that the projection $\pi_i(V)$ is a point,
and we may replace $V$ with the fibre of this projection and omit $W_i$. So
we have $\bdl(\a_i)=\dimo(\a_i)=d$.
Now let $\epsilon>0$ and $\tau \in \N$. Pick $Y_i \subset W_i(K)$ definable
over $\emptyset$ with $\a_i \in Y_i$ and $d \leq \bdl(Y_i) < d+\epsilon$.
Replacing $Y_i$ by $Y_i \cap X_i$ we may assume $Y_i$ is $cgp$ in $W_i$. Then
$\bdl(V \cap \prod_i Y_i) \geq \bdl(\a) = \dim V$. Let $Y_{i,s} := Y_i^{K_s}$
be the interpretation in $K_s$.
Then for $\U$-many $s$, for all $i \in \{1,\ldots ,n\}$, $Y_{i,s}$ is
$\tau$-$cgp$ in $W_i$, $1/\epsilon < |Y_{i,s}| < \infty$, and
$|V \cap \prod_i Y_{i,s}| \geq |Y_{i,s}|^\frac{\dim V}{d+2\epsilon}$. Hence
$V$ admits no power-saving.
\end{proof}
\subsection{Sharpness}
\label{subsect:sharp}
In this subsection, we show the converse to
Proposition~\ref{prop:coherentSpecial} and prove that every special subvariety
has no power saving. For this we will need to construct certain well chosen
cartesian products of finite sets, which are adapted to the special
subvariety.
The construction we are about to describe consists in building certain long
multi-dimensional arithmetic progressions on few algebraically independent
elements. The difficulty is that in order to belong to a given special
subgroup, these progressions will need to satisfy some almost invariance under
the division ring $F$ of $End^0(G)$ used to define the special subgroup. For
this purpose it will be convenient to introduce the notion of constrainedly
filtered ring, as follows.
\begin{definition}
A \defnstyle{constrained filtration} of a ring $\O$ is a chain $\O_n \subset
\O$ of finite subsets, $n \in \N$, such that
\begin{enumerate}
\item[(CF0)] $\bigcup_{n \in \N} \O_n = \O$, and $\forall n \in \N.\; \O_n
\subset \O_{n+1}$; \item[(CF1)] $\exists k \in \N.\; \forall n \in \N.\;
\O_n+\O_n \subset \O_{n+k}$;
\item[(CF2)] $\forall a \in \O.\; \exists k \in \N.\; \forall n \in \N.\;
a\O_n \subset \O_{n+k}$;
\item[(CF3)] $\forall \epsilon>0.\; \frac{|\O_{n+1}|}{|\O_n|} \leq
O_{\epsilon}(|\O_n|^\epsilon)$.
\end{enumerate}
If a constrained filtration exists, we say $\O$ is \defnstyle{constrainedly
filtered}.
\end{definition}
\begin{example}\label{eg:Zconstrained}
$\Z$ is constrainedly filtered, since $([-2^n,2^n])_n$ is a constrained
filtration.
\end{example}
Constrained filtrations are somewhat similar to Bourgain systems.
%Lemma{lem:QconstFilt}:
% $\Q$ is constrainedly filtered.
% .
%Proof:
% Let $\O_n := \{ a/n! : a \in \Z, |a/n!| \leq 2^n \}$.
%
% Then (CF0)-(CF2) are easily verified.
% For (CF3), note that for $\epsilon > 0$, we have $\frac{|\O_{n+1}|}{|\O_n|}
% = 2(n+1) \leq O(2^{n\epsilon}) \leq O(|\O_n|^\epsilon)$.
% .
\begin{lemma} \label{lem:constFilt}
Suppose $\O$ is a constrainedly filtered ring.
\begin{enumerate}[(i)]\item The polynomial ring $\O[X]$ is constrainedly
filtered.
\item If $\O$ is an integral domain and $a \in \O$, then the subring
$\O[a^{-1}]$ of the fraction field of $\O$ is constrainedly filtered.
\item If $\O' \supset \O$ is an extension ring in which $\O$ is central and
which is free and finitely generated as a $\O$-module,
then $\O'$ is constrainedly filtered.
\end{enumerate}
\end{lemma}
\begin{proof}
Say $(\O_n)_n$ is a constrained filtration of $\O$.
\begin{enumerate}[(i)]\item
Let $\O_n' := \sum_{i0$, for $n >> 0$,
$|\O_{n+1}'|/|\O_n'| = |\O_n|(|\O_{n+1}|/|\O_n|)^{n+1}
\leq O_\epsilon(|\O_n|^{1+(1+n)\epsilon})
\leq O_\epsilon(|\O_n'|^{\frac 1n + \frac{1+n}n \epsilon})
\leq O_\epsilon(|\O_n'|^{2\epsilon})$.
\item
Say $k \in \N$ is such that for all $n \in \N$ we have $a\O_n \subset
\O_{n+k}$ and $\O_n + \O_n \subset \O_{n+k}$.
Let $\O'_n := a^{-n}\O_{2kn}$.
Then
$\O'_n + \O'_n
= a^{-n}(\O_{2kn}+\O_{2kn})
\subset a^{-n}(\O_{2kn+k})
= a^{-(n+1)}(a\O_{2kn+k})
\subset a^{-(n+1)}(\O_{2kn+k+k})
= \O'_{n+1}$, so (CF1) holds.
(CF0) and (CF2) are immediate,
and (CF3) holds since $|\O'_n| = |\O_{2kn}|$.
\item
Say $\O' = \bigoplus_{i=1}^d a_i \O$.
Then let $\O'_n := \bigoplus_{i=1}^d a_i \O_n$.
Then (CF0), (CF1), and (CF3) are clear. For (CF2),
let $c_{ij}^t \in \O$ be such that $a_ia_j = \sum_t c_{ij}^t a_t$;
then given $\beta = \sum a_ib_i \in \O'$ with $b_i \in \O$,
let $k := \max_{i,j,t} (k_{b_i} + k_{c_{ij}^t})$
where $\alpha \O_n \subset \O_{n+k_\alpha}$ ($\forall n$),
and say $\O_n+\O_n \subset \O_{n+l}$ ($\forall n$).
Then $\beta\O'_n
= \sum_j \beta a_j \O_n
= \sum_{i,j} a_ib_ia_j \O_n
= \sum_{i,j} b_ia_ia_j \O_n
= \sum_{i,j,t} b_i c_{ij}^t a_t \O_n
\subset \sum_{i,j,t} a_t \O_{n+k}
\subset \O'_{n+k+d^2l}.$
\end{enumerate}
\end{proof}
%Lemma{lem:constFiltNoether}:
% Let $\O$ be a finitely generated integral domain extension of $\Q$,
% and $F$ its fraction field.
% Then there exists a constrainedly filtered ring $\O'$ which is a finitely
% generated ring extension of $\O$ and $\O \subset \O' \subset F$.
% .
%Proof:
% By Noether normalisation,
% $\O = \Q[\x][\tuple{\beta}]$
% where $\x \in \O^n$ is a transcendence basis for $F$ over $\Q$
% and $\beta_i \in \O$ is integral over $\Q[\x]$.
% By the primitive element theorem,
% $F = \Q(\x)[\alpha]$ for some $\alpha \in F$.
% So say $f \in \Q[\x]$ is such that, setting $\O'_0 := \Q[\x][1/f]$,
% the coefficients of the monic minimal polynomial for $\alpha$ over $\Q(\x)$
% are in $\O'_0$
% and $\beta_i \in \O'_0[\alpha] =: \O'$.
% Then $\O'_0[\alpha]$ is freely generated over $\O'_0$ by
% $1,\alpha,\ldots ,\alpha^{\deg(\alpha)-1}$.
%
% Now $\Q[\x]$ is constrainedly filtered by Lemmas~\ref{lem:QconstFilt} and
% \ref{lem:constFilt}(i),
% hence $\O'_0$ is constrainedly filtered by Lemma~\ref{lem:constFilt}(iii),
% hence $\O'$ is constrainedly filtered by Lemma~\ref{lem:constFilt}(iv).
%
% Now $\O \subset \O' \subset F$, as required.
% .
\begin{lemma} \label{lem:finDimConstFilt}
Suppose $D$ is a finite-dimensional algebra over a characteristic 0 field
$L$, and $\O \subset D$ is a finitely generated subring. Then there exists a
constrainedly filtered subring $\O' \subset D$ extending $\O$.
\end{lemma}
\begin{proof} Let $(e_k)_{1\leq k \leq d}$ be an $L$-basis of $D$ and
$(f_j)_j$ generators of $\O$. Without loss of generality we may change $L$
into the subfield generated by the co-ordinates $f_{j}^k$ of the $f_j$'s and
the co-ordinates $c_{ij}^k$ of the products $e_ie_j$'s. So we may assume
that $L$ is finitely generated. Let $\z=(z_1,\ldots,z_n)$ be a transcendence
basis for $L$ over $\Q$. Then $[L:\Q(\z)]$ is finite, so $D$ is again
finite-dimensional over $\Q(\z)$, and without loss of generality, we may
assume that $L=\Q(\z)$. There is a polynomial $g \in \Z[\z]$ such that all
$f_j^k$ and $c_{ij}^k$ belong to $\Z[\z,\frac{1}{g}]$. By
Example~\ref{eg:Zconstrained}, $\Z$ is constrainedly filtered. Then by
Lemma~\ref{lem:constFilt}(i), so is $\Z[\z]$, and by item $(ii)$ so is
$R:=\Z[\z,\frac{1}{g}]$, and by item $(iii)$ so is $\O' := \sum_{k=1}^d R
e_k \supset \O$.
% Let $R \subset \Q(\xitup)$ be a finitely generated subring extending
% $\Z[\xitup]$ such that $c_{ij}^k \in R$ and $\O \subset \O' := \sum_k
% Rb_k$.
% Now $\Z[\xitup]$ is constrainedly filtered by Lemma~\ref{lem:constFilt}(i),
% so $R$ is constrainedly filtered by Lemma~\ref{lem:constFilt}(ii),
% so $\O'$ is constrainedly filtered by Lemma~\ref{lem:constFilt}(iii).
% (The same proof does work in characteristic $p$, replacing $\Q$ with
% $\mathbb{F}_p$)
\end{proof}
\begin{fact} \label{fact:KA}
A division subring of a matrix algebra over a division ring has finite
dimension over its centre.
\end{fact}
\begin{proof}
This is a special case of the Kaplansky-Amitsur theorem
\cite[p17]{Jacobson-PIAlgs}, which shows that any primitive algebra
satisfying a proper polynomial identity is finite dimensional over its
centre. Indeed, any division ring is a primitive algebra, and any matrix
algebra $M_n(\Delta)$ over a division ring $\Delta$ satisfies a polynomial
identity (e.g. by the Amitsur-Levitzki theorem \cite[p21]{Jacobson-PIAlgs}).
% $M_n(\Delta)$ satisfies the the standard polynomial identity
% $S_{2n}(\x) = \sum_{\sigma \in S_{2n}} (\operatorname{sign} \sigma) \prod_i
% x_{\sigma i} = 0$.
% In fact, it is sufficient for present purposes to observe the rather easier
% fact that $M_n(\Delta)$ satisfies $S_{n^2+1}(\x) = 0$; this holds for
% dimension-theoretic reasons (\cite[p14]{Jacobson-PIAlgs}).
\end{proof}
In particular, combining this fact with the previous lemma, we see that if $F$
is a division subring of a matrix algebra, then every finite subset of $F$ is
contained in a constrainedly filtered subring of $F$. We will use this
observation in the next proposition. Although this is sufficient for our
purposes, we do not know it to be the optimal result along these lines - in
fact, for all we know, it could be that every finitely generated subring of
$M_n(\C)$ is constrainedly filtered.
\begin{proposition} \label{prop:converse}
Suppose $V \subset \prod_i W_i$ is special.
Then, for appropriate choices of $C_0 \leq \C$ and structures $K_s$ with
universe $\C$ and scaling constant $\xi \in \stR$,
the variety $V$ has a coherent generic $\a \in V(K)$ such that each $a_i$ is
$dcgp$ in $W_i$.
\end{proposition}
\begin{proof}
The conclusion is preserved by taking products and under correspondences, so
we may assume $W_i = G$ where $G$ is a $d$-dimensional commutative connected
algebraic group defined over a countable algebraically closed subfield $C_0
\subset \C$,
and $V = H \leq G^n$ is a special subgroup.
By Remark~\ref{remk:lieSpecial} and permuting co-ordinates, we may assume
that the Lie subalgebra
$\Lie(H) \leq \Lie(G)^n$ is the graph of an $F$-linear map $\theta =
(\theta_1,\ldots ,\theta_m) : \Lie(G)^m \rightarrow \Lie(G)^{n-m}$ with
$\theta_i(\x) = \sum_{j=1}^d\alpha_{ij}x_j$, where $\alpha_{ij} \in F$ and
$F$ is a division subring of $\End^0(G):=\End(G) \otimes_{\Z} \Q$. We may
assume that $F$ is generated by the $\alpha_{ij}$.
Extending $C_0$ if necessary, we may assume $F \subset \End^0_{C_0}(G)$,
i.e.\ that every element of $F$ is a scalar multiple of an algebraic
endomorphism of $G$ which is defined over $C_0$.
Now $F$ acts faithfully by $\C$-linear maps on $\Lie(G) \cong \C^d$, so by
Fact~\ref{fact:KA}, $F$ has finite dimension over its centre.
So by Lemma~\ref{lem:finDimConstFilt}, there is a constrainedly filtered
subring $\O \subset F$ such that $\alpha_{ij} \in \O$ ($\forall i,j$).
% Since $\O$ is countable, without loss of generality we may enlarge $C_0$ if
% necessary and assume that all elements in $\O$ are scalar multiple of
% endomorphisms of $G$, which are defined over $C_0$.
% \mbcomment{But it isn't immediate that $\O$ is countable if $F$ isn't, so
% better to first shrink $F$ to be countable}.
Say $(\O_n)_n$ is a constrained filtration for $\O$. Let $\exp_G : \Lie(G)
\twoheadrightarrow G(\C)$ be the Lie exponential map, which is a surjective
$\End(G)$-homomorphism.
%(Readers averse to Lie theory should see Remark~\ref{remk:profiniteCovers}
%below for a purely algebraic substitute.)
For every positive integer $s$, let $\gammatup_s =
(\gamma_{s,i})_{i=1,\ldots ,s} \in G(\C)^s$ be algebraically generic over
$C_0$, i.e.\ $\dim^0(\gammatup_s) = sd$,
and let $\gammatup'_s \in \Lie(G)^s$ be arbitrary such that
$\exp_G(\gamma'_{s,i}) = \gamma_{s,i}$.
Note then that $\gammatup'_s$ is $F$-linearly independent modulo
$\ker(\exp_G)$.
For $k \in \N_{\ge 0}$ we set $X_{k,s} := \sum_{i=1}^s\O_{s-k}\gamma'_{s,i}
\subset \Lie(G)$ if $s\geq k$ and $X_{k,s} := \emptyset $ if $s0}$, we have
$\frac{|X_{k,s}|}{|X_{k+1,s}|} = \frac{|\O_{s-k}|^s}{|\O_{s-k-1}|^s} \leq
O_\epsilon(|\O_{s-k-1}|^{s\epsilon}) = O_\epsilon(|X_{k+1,s}|^\epsilon)$.
Hence $\bdl(X_k) \leq (1+\epsilon)\bdl(X_{k+1})$ for any $\epsilon>0$,
so $\bdl(X_k) \leq \bdl(X_{k+1})$.
Clearly $\bdl(X_{k+1}) \leq \bdl(X_k)$,
so $\bdl(X_{k+1}) = \bdl(X_k)$. So by induction, $\bdl(X_k) = d$ for all
$k$, so $\bdl(Z) = \inf_k \bdl(X_k) = d$.
\end{proof}
Now since $Z$ is an $\O$-submodule, the co-ordinate projection to $\Lie(G)^m$
induces a bijection from $\Lie(H)
\cap Z^n$ to $Z^m$, so $\bdl(\Lie(H) \cap Z^n) = md$. Moreover $\exp_G$ is
injective on each $X_{k,s}$ and hence on each $X_k$ and on $Z$.
\begin{claim}
$\exp_G(X_0)$ is $cgp$ in $G$.
\end{claim}
\begin{proof}
Suppose $W \subsetneq G$ is a proper closed subvariety over $K$.
Say $C_0(\b) \subset K$ is a finitely generated extension of $C_0$ such
that $W$ is over $C_0(\b)$. Then $W = J_{\b}$ for some $J_{\x}$ a
constructible family defined over $C_0$ of proper closed subvarieties of
$G$. If $\b = \lim_{s \rightarrow \U} \b_s$ and $W_s := J_{\b_s}$ then
$W(K) = \prod_{s \rightarrow \U} W_s(\C)$.
It holds for $\U$-many $s$ that $\dim^0(\b_s) \leq \dim^0(\b) =: k$.
We claim that for such $s$, we have $\operatorname{rk}_F (
\exp_G^{-1}(W_s(\C)) \cap X_{0,s} ) \leq k$.
Indeed, suppose $\g' = (g_i)_{i=0}^k$ is $F$-linearly independent with
$g_i' \in \exp_G^{-1}(W_s(\C)) \cap X_{0,s}$. By $F$-linear algebra, some
$k+1$-subtuple $\gammatup''$ of the generators $\gammatup'_s$ of $X_{0,s}$
is in the $F$-linear span of $\g'$. Let $g_i := \exp_G(g_i')$. Then
$\dim^0(\g) \geq \dim^0(\exp_G(\gammatup'')) = (k+1)d$,
so $\dim^0(\g) = (k+1)d$.
Then $\dim^0(\g/\b_s) \leq \dim(W_s^{k+1}) \leq (k+1)(d-1) = \dim^0(\g) -
(k+1) < \dim^0(\g) - \dim^0(\b_s)$,
so $\dim^0(\b_s) < \dim^0(\g) - \dim^0(\g/\b_s) = \dim^0(\b_s) -
\dim^0(\b_s/\g) \leq \dim^0(\b_s)$, which is a contradiction.
So $| W_s(\C) \cap \exp_G(X_{0,s}) | = | \exp_G^{-1}(W_s(\C)) \cap X_{0,s}
| \leq | \O_s |^k = (|\O_s|^s)^{\frac ks} = |X_{0,s}|^{\frac ks}$.
So for any $\epsilon \in \R_{>0}$, considering sufficiently large $s$, we
deduce $\bdl(W(K) \cap \exp_G(X_0)) \leq \epsilon\bdl(X_0)$.
Hence $\bdl(W(K) \cap \exp_G(X_0)) = 0$, as required.
\end{proof}
% True, but not needed:
% Moreover, $\exp_G$ induces a bijection $LH \cap Z^n \rightarrow H \cap
% \exp(Z)^n$;
% indeed, if $\exp(\z,\z') \in H$ then $(\z,\z') \in LH + (\ker\exp_G)^n$,
% so say $\z'+\zeta' = \theta(\z+\zeta) = \theta(\z)+\theta(\zeta)$,
% so since $Z \cap \ker\exp_G = \emptyset $, already $\theta(\z)=\z'$, so
% $(\z,\z') \in LH \cap Z^n$.
By Fact \ref{fact:ideal} we can pick $\a \in H \cap \exp_G(Z)^n$ with
$\bdl(\a) = \bdl(H \cap \exp_G(Z)^n)$. By injectivity of $\exp_G$ on $Z$
this is $\geq \bdl (\Lie(H) \cap Z^n) = md$.
Note that $\bdl(a_i) \leq \bdl(Z)=d$ and that, by the above Claim $a_i \in
\exp_G(X_0)$ is $dcgp$ in $G$ (see Def. \ref{def-dcgp}) and hence $cgp$ (see
Lemma \ref{cgplem}). So by Lemma~\ref{lem1} $\a$ is coherent.
\end{proof}
\begin{remark} \label{remk:profiniteCovers}
The only essential role played by Lie theory in the above proof is to
establish that $F$ embeds in a matrix algebra; for the rest of the proof,
$\exp_G : \Lie(G) \rightarrow G$ is used only to pick out choices of systems
of division points of elements of $G$, and this can instead be done directly
by replacing $\exp_G$ with $\rho : \widehat G \rightarrow G$ where $\widehat
G$ is the ``profinite cover'' $\widehat G := \invlim ([n] : G \rightarrow
G)$ consisting of ``division systems'' $(x_n)_n$ satisfying $[n]x_{nm} =
x_m$, and $\rho$ is the first co-ordinate map of the inverse limit,
$\rho((x_n)_n) := x_1$. Then $\End^0(G)$ acts on $\widehat G$ by $\frac\eta
m (x_n)_n := (\eta x_{nm})_n$.
\end{remark}
\begin{remark} \label{remk:converseGp}
In the case that $G$ is a semiabelian variety, we can do slightly better and
obtain approximate $\O$-modules which are in general position in the sense
of Elekes-Szab\'o rather than merely in coarse general position. More
precisely, say an internal subset $X \subset G(K)$ is in general position if
it has finite intersection with any proper closed subvariety $W \subsetneq
G$ over $K$. Then proceed as in the proof of Proposition~\ref{prop:converse}
but taking $\gammatup_s := \gamma \in G(\C)$ to be a singleton which is in
no proper algebraic subgroup of $G$. Let $\Gamma := \O\gamma \leq G(\C)$ be
the $\O$-submodule generated by $\gamma$, which is a finitely generated
subgroup of $G(\C)$. As shown in
\cite[Theorem~4.7]{Scanlon-automaticUniformity},
as a consequence of the truth of the Mordell-Lang conjecture,
if $V_b$ is a constructible family of subvarieties of $G$ then there is a
uniform bound on the sizes of finite intersections $V_b \cap \Gamma$.
Hence $\exp_G(X_k)$ is in general position in $G$.
However, this approach clearly fails for $G=\mathbb{G}_a^2$, by considering
intersections with linear subvarieties. Pach
\cite[Theorem~2]{Pach-midpoints} gives an example of an internal subset $X
\subset K^2$ with $\bdl(X)=1=\bdl(X+X)$ and where the intersection with any
linear subvariety has size at most 2; however, quadratic subvarieties
witness that this $X$ is not in general position. We do not know whether it
is possible to find such an $X$ which is in general position. This prompts
the following question.
\end{remark}
\begin{question}
Is there a sequence of finite sets $A_n \subset \C^2$ such that $|A_n|\to
+\infty$, $|A_n + A_n| \leq |A_n|^{1 + 1/n}$ and with the property that,
for each degree $d$, $$\sup_{n, \mathcal{C}, \deg \mathcal{C} \leq d} |A_n
\cap \mathcal{C}|$$ is finite, where $\mathcal{C}$ runs through algebraic
curves $\mathcal{C} \subset \C^2$.
\end{question}
\subsection{Proofs of the main results}
\label{subsect:proofs}
We first observe that Theorem \ref{thm:main1} is a special case of Theorem
\ref{thm:main}, as follows immediately from the following lemma.
\begin{lemma}\label{lem:1dimSpecial}
Let $G$ be a 1-dimensional connected complex algebraic group, let $n \geq
1$, and let $H \leq G^n$ be a connected algebraic subgroup. Then $H$
is a special subgroup of $G^n$.
\end{lemma}
\begin{proof}
First, suppose $G$ is the additive group $(\C;+)$.
Then $F := \End^0(G) = \C$ is a division ring, and $H$ is a vector subgroup,
i.e.\ $H = \ker(A)$ for some $A \in \Mat_n(\C)$, as required.
Otherwise, $G$ is the multiplicative group or an elliptic curve. In either
case, $F := \End^0(G)$ is again a division ring, being either $\Q$ or a
quadratic imaginary field extension of $\Q$.
We refer to \cite[Lemma~4.1(i)]{coversfRM} for the fact that $H \leq G^n$ is
the connected component of $\ker(A)$ for some $A \in \Mat_n(\End(G))$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:main}]
Combine Lemma~\ref{lem:powersavingCoherent} with
Proposition~\ref{prop:coherentSpecial} and Proposition~\ref{prop:converse}.
\end{proof}
We end this section with a proof of Corollary \ref{gen-sum-product} from the
Introduction. It is a special case of the following more precise result:
\begin{corollary}
Suppose $(G_1;\cdot_1),(G_2;\cdot_2)$ are non-isogenous connected complex
algebraic groups of the same dimension,
and $\Gamma \subset G_1 \times G_2$ is a generically finite algebraic
correspondence.
Then there are $\tau,\epsilon,c>0$ such that if $A_i \subset G_i$ are finite
sets such that $\Gamma \cap (A_1 \times A_2)$ is the graph of a bijection
between $A_1$ and $A_2$,
and if $A_i \subset G_i$ and $A_i \cdot_i A_i \subset G_i$ are $\tau$-$cgp$
for $i=1,2$,
then
\[ \max( |A_1 \cdot_1 A_1|, |A_2 \cdot_2 A_2| ) \geq c|A_1|^{1+\epsilon} .\]
\end{corollary}
\begin{remark}
The $cgp$ condition holds trivially for any $A$ when $\dim(G_i)=1$.
% The usual sum-product phenomenon is the case $G_1 = (\C;+)$ and $G_2 =
% (\C^\times;\cdot)$ and $\Gamma = \{ (x,x) : x \in \C^\times \}$.
%
% The case of the additive group and an elliptic curve was considered for
% finite fields in \cite{shparlinski-sumProdElliptic}; this corresponds to
% the case of the corollary with $G_1 = (\C;+)$ and $G_2 = E_\lambda \subset
% \P^2(\C)$ defined by $y^2=x(x-1)(x-\lambda)$ with $\lambda \in \C \setminus
% \{0,1\}$, and $\Gamma = \{ (x,[x:y:1]) : [x:y:1] \in E_\lambda \}$.
\end{remark}
\begin{proof}
Let $V = \{ (x_1,y_1,x_1\cdot_1y_1,x_2,y_2,x_2\cdot_2y_2) :
(x_1,x_2),(y_1,y_2) \in \Gamma \}$.
Suppose $V$ is special.
Then $V$ is in co-ordinatewise correspondence with a special subgroup $H
\leq G^6$, say. As in Remark~\ref{rmk:mainCodim1}, the projection of $H$ to
the first three co-ordinates is in co-ordinatewise correspondence with the
graph $\{ (x,y,x+y) \}$ of the group operation of $G$. Hence the graph of
the group operation of $G_1$ is in co-ordinatewise correspondence with that
of $G$.
By Fact~\ref{fact:corrIsog}, $G_1$ is commutative and isogenous to $G$.
Similarly, considering the projection to the last three co-ordinates, $G_2$
is commutative and isogenous to $G$. Since isogeny is an equivalence
relation, this contradicts the assumption that $G_1$ and $G_2$ are not
isogenous.
So by Theorem~\ref{thm:main}, $V$ admits a power-saving, say by $\epsilon'$.
So for sufficiently large $\tau$, if $A_i$ are as in the statement, then
setting $X := \{ (a_1,b_1, a_1\cdot_1 b_1, a_2,b_2, a_2\cdot_2 b_2) :
a_i,b_i \in A_i; (a_1,a_2),(b_1,b_2) \in \Gamma \} \subset V$, we have
$|A_1|^2 = |X| \leq O(\max( |A_1 \cdot_1 A_1|, |A_2 \cdot_2 A_2|
)^{2-\epsilon'})$.
So $\epsilon := \frac{\epsilon'}{2-\epsilon'}$ is as required.
\end{proof}
\begin{question}
Consider the following weakening of coarse general position. Say $a \in
K^{<\infty}$ with $\bdl(a)=\dimo(a)$ is in \emph{Larsen-Pink general
position} if for any $B \subset K^{<\infty}$, we have $\bdl(a/B) \leq
\dimo(a/B)$. This hypothesis suffices to give the trivial upper bound of
Lemma~\ref{lem:power-saving}, and it is not satisfied by the counterexample
of Section~\ref{gpNecessity}, nor by similar constructions based on
nilpotent groups.
Does Theorem~\ref{thm:main} go through unchanged if coarse general position
is relaxed Larsen-Pink general position?
The techniques of this paper are insufficient to prove this generalisation,
as the corresponding weakened notion of coherence does not directly yield a
pregeometry.
Since type-definable subgroups of simple algebraic groups do satisfy this
Larsen-Pink condition (see \cite[2.15]{Hr-psfDims}), a positive answer
should suffice to recover the characteristic 0 case of the main theorem of
\cite{bgt}, Theorem~5.5.
\end{question}
\section{Coherence in subgroups}\label{sec:subgroups}
In this section we observe a strengthening of our results in the special case
of a $\bigwedge$-definable pseudo-finite subgroup and derive Theorem
\ref{thm:coherentApproxSubgroup} from the introduction. We then briefly
discuss connections to Diophantine problems and Manin-Mumford.
%Here, we do not make Assumption~\ref{assumptionC}.
\begin{theorem} \label{thm:coherentSubgroup} We keep the notational setup of
Section \ref{sec:setup}.
Let $G$ be a commutative algebraic group over $C_0$ and $\Gamma \leq G(K)$
be a $\bigwedge$-definable (over $\emptyset $) subgroup of $G(K)$ contained
in a $cgp$ definable (over $\emptyset$) subset of $G$ (see
Definition~\ref{def-dcgp}). Assume $\bdl(\Gamma) = \dim(G)$. Then the locus
$\locus^{G^n}(\gammatup/C_0)$ of any coherent tuple $\gammatup \in \Gamma^n$
is a coset of an algebraic subgroup.
\end{theorem}
\begin{remark}
The commutative case is the only relevant case: by
Corollary~\ref{specialAbelian}, if $G$ is a connected algebraic group with
such a subgroup $\Gamma$ then $G$ is commutative.
% But because we want to apply this theorem to groups which aren't
% necessarily connected, and don't want to have to deal with
% not-quite-abelian groups in the proof, we restrict to commutative groups
% in the statement.
\end{remark}
\begin{proof}
By Lemma~\ref{cgplem} any $\alpha \in \Gamma$ is $cgp$. In particular if
$\bdl(\alpha)>0$, then $\alpha$ is generic in $G$ and $\bdl(\alpha) \leq
\bdl(\Gamma)=\dim(G)=\dimo(\alpha)$. So for all $\alpha \in \Gamma$ we have
$\bdl(\alpha) \leq \dimo(\alpha)$.
By Fact~\ref{fact:ideal}, we may find $\alphatup \in \Gamma^n$ with
$\bdl(\alphatup/\gammatup) = \bdl(\Gamma^n) = n\dim(G)$.
Then
$\bdl(\gammatup,\alphatup,\gammatup+\alphatup) = \bdl(\gammatup,\alphatup) =
\bdl(\gammatup) + n\dim(G) = \dimo(\gammatup) + n\dim(G) \geq
\dimo(\gammatup,\alphatup) =
\dimo(\gammatup,\alphatup,\gammatup+\alphatup)$,
and $\gamma_i, \alpha_i, \gamma_i+\alpha_i \in \Gamma$,
so $(\gammatup,\alphatup,\gammatup+\alphatup)$ is coherent by
Lemma~\ref{lem1}.
Since the product of cosets is a coset, we may assume that
$\locus^{G^n}(\gammatup/C_0)$ is not a product of loci of subtuples.
Then as in Proposition~\ref{prop:coherentSpecial},
there is a commutative algebraic group $G'$ over $C_0$
and a tuple $(\gammatup',\alphatup',\psitup')$ which is generic in a
connected subgroup $H \leq G'^{3n}$ and which is co-ordinatewise
$\acl^0$-interalgebraic with $(\gammatup,\alphatup,\gammatup+\alphatup)$.
Furthermore, for each $i$ we have $\theta_i^1\gamma_i' + \theta_i^2\alpha_i'
+ \theta_i^3\psi_i' \in G'(C_0)$ for some $\theta_i^j \in \End_{C_0}(G')$
invertible in $\End^0_{C_0}(G')$, and so we may assume without loss that
$\psi_i' = \gamma_i'+\alpha_i'$.
Then by Fact~\ref{fact:corrIsog}, there are $m \in \N$ and isogenies $\eta_i
: G' \rightarrow G$ and $k_i \in G(C_0)$ such that $\eta_i(\gamma'_i) =
m\gamma_i + k_i$ for $i=1,\ldots ,n$.
Hence $\locus^{G^n}(m\gammatup/C_0) =
(\prod_i\eta_i)(\locus^{G^n}(\gammatup')) - \k = (\prod_i\eta_i)(\pi_1(H)) -
\k$ where $\k=(k_1,\ldots ,k_n)$ and $\pi_1 : (\x,\y,\z) \mapsto \x$.
So $\locus^{G^n}(m\gammatup/C_0)$ is a coset of an algebraic subgroup of
$G^n$,
and hence so is $\locus^{G^n}(\gammatup/C_0)$, as required.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:coherentApproxSubgroup}]
First, note that there exists a function $f : \N \rightarrow \N$ such that
any translate of a subvariety of $G^n$ of complexity $\leq \tau$ has
complexity $\leq f(\tau)$.
This follows from \cite[Lemma~3.4]{bgt}, and can also be seen as a
consequence of the fact that the family of translates of subvarieties in a
constructible family is constructible.
Increasing $f$ if necessary, we may assume also that $f$ is strictly
increasing and $2^{-\tau} + \frac1{f(\tau)} \leq \frac1\tau$ for any $\tau
\in \N$.
By \cite[Proposition~2.26]{TaoVu}, there are $C_1,C_2>0$ such that if $|A+A|
\leq |A|^{1+\eps}$ then $H := A-A$ is an $|A|^{C_1\eps}$-approximate
subgroup and $|H| \leq |A|^{1+C_2\eps}$.
It therefore suffices to prove the following revised statement:
there are $N,\tau,\eps,\eta>0$ depending only on $G$ and the complexity of
$V$ such that if $H \subset G$ is a $\tau$-$cgp$ $|H|^\eps$-approximate
subgroup and $|H| \geq N$, then $|H^n \cap V| < |H|^{\frac{\dim(V)}{\dim(G)}
- \eta}$.
Indeed, given $\tau' \in \N$, if $(N,\tau,\eps,\eta)$ are as required in the
revised statement for $V$ of complexity at most $f(\tau')$, then
$(N,\tau,\eps':=\frac{\eps}C_1,\eta':=\frac{\dim(V)}{\dim(G)} -
(\frac{\dim(V)}{\dim(G)} - \eta)(1 + C_2\eps'))$
are as required in the original statement for $V$ of complexity at most
$\tau'$, after increasing $C_1$ if necessary to ensure $\eta'>0$.
Indeed, given $V$ of complexity at most $\tau'$, suppose there is $A \subset
G$ with $|A| \geq N$ and $|A^n \cap V| \geq |A|^{\frac{\dim(V)}{\dim(G)} -
\eta'}$ and $|A+A| \leq |A|^{1+\eps'}$,
and with $H := A-A$ being $\tau$-$cgp$.
Then $H$ is an $|A|^{C_1\eps'} = |A|^{\eps}$-approximate subgroup, and $|H|
\geq |A| \geq N$.
Let $x \in A$, and let $V' := V-(x,x,\ldots ,x)$, which has complexity at
most $f(\tau')$.
Then $|H^n \cap V'| \geq |A^n \cap V|
\geq |A|^{\frac{\dim(V)}{\dim(G)} - \eta'}
\geq |H|^{(\frac{\dim(V)}{\dim(G)} - \eta')(1+C_2\eps')^{-1}}
\geq |H|^{\frac{\dim(V)}{\dim(G)} - \eta}$.
This contradicts the revised statement.
Now suppose the revised statement fails. Then there is a family $(V_b)_b$ of
bounded complexity subvarieties of $G^n$, such that for each $s \in \N$
there is an $f(s)$-$cgp$ subset $H_s \subset G$ such that $H_s$ is an
$|H_s|^{4^{-s}}$-approximate subgroup, and a parameter $b_s$, such that
$|H_s| \geq s$ and $|H_s^n \cap V_{b_s}| \geq
|H_s|^{\frac{\dim(V_{b_s})}{\dim(G)} - \frac1s}$ and $V_{b_s}$ is not a
coset.
Let $\U$ be a non-principal ultrafilter on $\N$,
let $\Gamma_i := \prod_{s \rightarrow \U} \sum_{j=1}^{2^{s-i}}H_s$ for $i
\in \N$,
and let $\Gamma := \bigcap_{i\geq 0} \Gamma_i$.
Set the language $\L$ to be such that each $\Gamma_i$ is definable.
Then $\Gamma$ is a $\bigwedge$-definable subgroup,
since $\Gamma_{i+1} + \Gamma_{i+1} \subset \Gamma_i$.
Now $\bdl(\Gamma_i) = \bdl(\prod_{s \rightarrow \U} H_s)$ for any $i$,
because $|\sum_{j=1}^{2^{s-i}}H_s| \leq |H_s|^{1+2^{s-i}4^{-s}} \leq
|H_s|^{1+2^{-s}}$.
So $\bdl(\Gamma) = \bdl(\prod_{s \rightarrow \U} H_s)$,
so setting our scaling parameter $\xi$ appropriately,
we may ensure $\bdl(\Gamma)=\dim(G)$.
We claim that $\Gamma_0$ is $cgp$ in $G$. Indeed, $\sum_{j=1}^{2^s}H_s$ is
contained in the union of $|H_s|^{2^s 4^{-s}} = |H_s|^{2^{-s}}$ translates
of $H_s$,
so if $W \subsetneq G$ has complexity $\leq s$ then, using the lower bound
we assumed on $f$, we have
$|W \cap \sum_{j=1}^{2^s}H_s| \leq |H_s|^{2^{-s}} |H_s|^{\frac1{f(s)}} \leq
|H_s|^{\frac1s}$.
So $\sum_{j=1}^{2^s}H_s$ is $s$-$cgp$ in $G$.
% By Lemma~\ref{lem:cgpComparison}, it follows that $\Gamma$ is dcgp.
Let $b := \lim_{s \rightarrow \U} b_s$.
Then $\bdl(\Gamma^n \cap V_b) = \dim(V_b)$.
Set $C_0$ such that $G$ and $V_b$ are over $C_0$.
By Fact \ref{fact:ideal} we can pick $\gammatup=(\gamma_1,\ldots,\gamma_n)
\in \Gamma^n\cap V_b$ with $\bdl(\gammatup) = \bdl(\Gamma^n \cap V_b) $.
Since $\Gamma_0$ is $cgp$ in $G$, Lemma \ref{cgplem} implies that all
$\gamma_i$'s are $cgp$ and $\bdl(\gamma_i)\leq \dimo(\gamma_i)$. Then
$\gammatup$ is coherent by Lemma~\ref{lem1}. So by
Theorem~\ref{thm:coherentSubgroup} we have that $V_b =
\locus^{G^n}(\gammatup/C_0)$ is a coset.
But then so is $V_{b_m}$ for $\U$-many $m$, contradicting our assumption.
\end{proof}
\begin{example}[Connections with Manin-Mumford and Mordell-Lang]\label{MM}
Let $G$ be a complex elliptic curve.
Write $G[\infty] := \bigcup_{r \in \N} G[r]$ for the torsion subgroup.
Suppose $V \subset G^n$ is an irreducible closed complex subvariety such
that $V(\C) \cap G[\infty]$ is Zariski dense in $V$.
We know, by the Manin-Mumford conjecture proven by Raynaud, that $V$ is a
coset of an algebraic subgroup. Some co-ordinate projection to
$G^{\dim(V)}$, yields an isogeny, so $V = H+\alpha$ where
$H=\eta(G^{\dim(V)})$ is a subgroup and $\eta$ is an isogeny, and $\alpha
\in G[\infty]^n$.
Setting $c := |\ker(\eta)|^{-1}$
it follows that for $r \geq N:=\operatorname{ord}(\alpha)$ we have $|V(\C)
\cap G[r!]^n| \geq c\cdot|G^{\dim(V)}[r!]|$, and so for $r \in \N$ we have
the lower bound $|V(\C) \cap G[r!]^n| \geq \Omega(|G[r!]|^{\dim(V)})$.
Suppose conversely that we only know this consequence of Manin-Mumford on
the asymptotics of the number of torsion points in $V$, or even just that
for every $\epsilon > 0$, for arbitrarily large $r \in \N$ we have $|V(\C)
\cap G[r!]^n| \geq |G[r!]|^{{\dim(V)}-\epsilon}$.
Then it follows that $V$ is a coset of an algebraic subgroup.
Indeed, $G[r!]$ is a subgroup and is trivially $\tau$-$cgp$ for any $\tau$
since $\dim(G)=1$, so this is an immediate consequence of
Theorem~\ref{thm:coherentApproxSubgroup}.
We can generalise this argument by replacing $G[\infty]$ with a finite rank
subgroup, as in the Mordell-Lang conjecture. Indeed, let $\Gamma \leq G(\C)$
be a finite rank $\End(G)$-submodule.
Say $\Gamma$ is contained in the divisible hull of the subgroup generated by
$\gamma_1,\ldots ,\gamma_k$.
Let $\Gamma_r := \{ x \in \Gamma : (r!)x \in \sum_i [-r,\ldots ,r]\gamma_i
\}$.
Then $\Gamma_r$ is finite and $|\Gamma_r+\Gamma_r|\leq 2^k|\Gamma_r|$, so as
above we obtain that if $V \subset G^n$ is an irreducible closed subvariety,
then $V$ is a coset of a subgroup if and only if for all $\epsilon>0$, for
arbitrarily large $r \in \N$, we have $|V(\C) \cap \Gamma_r^n| \geq
|\Gamma_r|^{\dim(V)-\epsilon}$.
\end{example}
% Note can do the same for simple abelian varieties, if we also allow
% ourselves enough of MM to see that \Gamma is still cgp (enough to use
% consequences of MM on asymptotics of torsion points on subvarieties of A)
% Probably we can do something similar for Zhang's theorem?
\appendix
\section{Projective geometries fully embedded in algebraic geometry}
\label{appx:EHeq}
\cite{EH-projACF} characterises the projective subgeometries of the geometry
of algebraic closure in an algebraically closed field $K$ over an
algebraically closed subfield $C_0$. The points of such a geometry are
$C_0$-interalgebraicity classes of elements of $K$.
In this essentially self-contained appendix, we consider the more general
situation of a projective geometry induced from field-theoretic algebraic
dependence whose points are $C_0$-interalgebraicity classes of finite tuples
from $K$ (or, equivalently, of $K$-rational points of arbitrary varieties over
$K$).
The arguments are generalisations of those used in \cite{EH-projACF}. We use
Hrushovski's abelian group configuration theorem to find an abelian algebraic
group, then apply a version of the fundamental theorem of projective geometry
to identify the co-ordinatising skew field of the geometry as a skew field of
quasi-isogenies of the group. Identifying the isogenies involved requires a
little more care in the higher-dimensional case, as there may be non-trivial
endomorphisms which are not isogenies, and these cannot appear in the
co-ordinatising skew field.
We allow ourselves to simplify some of the algebra by restricting ourselves to
the characteristic 0 case, whereas \cite{EH-projACF} works in arbitrary
characteristic.
%We leave determining the correct positive characteristic version to the
%future.
Let $K$ be an algebraically closed field of characteristic 0.
%We work with $\Keq$. Those not familiar with this notion may take as the
%definition $\Keq := K^{<\omega}$, the set of finite tuples of elements of
%$K$, and in any case we sometimes identify an element of $\Keq$ with an
%interdefinable element of $K^{<\omega}$.
Let $C_0 \leq K$ be an algebraically closed subfield, and let $\cl :
\powerset(K^{<\infty}) \rightarrow \powerset(K^{<\infty})$ be field-theoretic
algebraic closure over $C_0$ as defined in Example \ref{alg-tuples}. In other
words, for a subset $B \subset K^{<\infty}:= \bigcup_{n \geq 1} K^n$ we let
$\cl(B)$ be the set of tuples from the field-theoretic algebraic closure
$C_0(B)^{\alg}$ of the subfield $C_0(B) \leq K$ generated by $C_0$ and the
co-ordinates of all tuples from $B$. This closure operator\footnote{In model
theory this is usually denoted by $\acleq$ and is defined on subsets of
$\Keq$, which we identify here with $K^{<\omega}$ via elimination of
imaginaries.
} was denoted $\acl^0(B)^{<\infty}$ in Section \ref{sec:proj}. So $a \in
\cl(B)$ if and only if $a$ has finite orbit under $\Aut(K/C_0(B))$, if and
only if $a \in (C_0(B)^{\alg})^{<\omega}$.
If $V$ is an algebraic variety over $C_0$ and $a \in V(K)$ is a $K$-rational
point, we may consider $a$ as a tuple in $K^{<\infty}$ as explained in \S
\ref{absVars}.
%Since in fact we work only up to $\acleq$-interalgebraicity, the reader may
%instead suppose that we fix a finite atlas for $V$ with charts affine
%varieties over $C_0$, then identify $a \in V(K)$ with its co-ordinates in a
%chart; this is well-defined up to $\acleq$.
Let $\G_K := \P(K^{<\infty};\cl)$ be the projectivisation of the closure
structure $(K^{<\infty};\cl)$, as defined in \textsection\ref{subsect:geoms};
i.e.\ $\G_K = \{ \cl(\{a\}) : a \in K^{< \infty} \setminus \cl(\emptyset ) \}$
with the closure induced from (and still denoted by) $\cl$.
For $x \in K^{< \infty}$ and $C \subset K^{< \infty}$, define $\widetilde x :=
\cl(\{x\})$
and $\widetilde C := \{ \widetilde c : c \in C \}$.
As already noted $\G_K$ is not a geometry in general (it does not satisfy the
exchange property), but here we are interested in geometries that embed in
$\G_K$. We say that a geometry $P$ is \emph{connected} if any two points $a,b$
are non-orthogonal, i.e. if there exists $C \subset P$ such that $a \in
\cl(b,C) \setminus \cl(C)$.
\begin{lemma} \label{lem:fullyEmbedded}
Let $P \subset \G_K$ and suppose that the restriction $(P,\cl)$ of $\cl$ to
$P$ forms a connected geometry (embedded in $\G_K$). Then the following are
equivalent:
\begin{enumerate}[(i)]\item for any $\widetilde x \in P$ and $\widetilde C
\subset P$,
$\widetilde x \in \cl(\widetilde C) \Leftrightarrow x \nind_{C_0} C$
% \item[(i')] for any $\widetilde x \in P$ and $\widetilde C \subset P$,
% $\widetilde x \in \cl(\widetilde C) \Leftrightarrow \trd(x/C_0(C)) = 0
% \Leftrightarrow \trd(x/C_0(C)) < \trd(x/C_0)$.
\item[(ii)]
There exists $k \in \N$ such that for any finite subset $\widetilde A
\subset P$,
\[ k \cdot \dim_{\cl}(\widetilde A) = \dimo(A),\]
where recall $\dimo(A):= \trd(C_0(A)/C_0)$.
\end{enumerate}
\end{lemma}
\begin{proof} Note that (i) is equivalent to say that $\dimo(x/C)=0$ if and
only if $\dimo(x/C)<\dimo(x)$.
That (i) implies (ii) follows from additivity of dimensions, setting $k :=
\dimo(a)$ for any $a \in K^{<\infty}$ such that $\widetilde a \in P$, once
we see that this does not depend on the choice of $a$. For another such $b$
with
$\widetilde b \in P$ we show that $\dimo(a)=\dimo(b)$. If $\widetilde b =
\widetilde a$ then $b$ and $a$ are interalgebraic over $C_0$, so this holds.
Else by non-orthogonality there is $\widetilde C$ such that $\widetilde a
\in \cl(\widetilde C \widetilde b)$ and $\widetilde b \in \cl(\widetilde C
\widetilde a)$ but $\widetilde a,\widetilde b \notin \cl(\widetilde C)$.
Then by (i), $\dimo(a) = \dimo(a/C) = \dimo(b/C) = \dimo(b)$.
The converse is easy and since it is not needed in the sequel, we leave it to
the reader.
\end{proof}
\begin{definition} \label{defn:fullyEmbedded}
We say a connected geometry $(P,\cl) \subset \G_K$ is
\defnstyle{($k$-dimensionally) fully embedded in $\G_K$} if the equivalent
conditions of the above lemma hold.
\end{definition}
If $G$ is a connected abelian algebraic group over $C_0$,
let $E_G := \End_{C_0}(G)$ be the ring of algebraic endomorphisms defined over
$C_0$,
and let $E^0_G := \End^0_{C_0}(G) := \Q \otimes_\Z E_G$.
Any $\eta \in E^0_G$ can be written as $q\eta'$ for some $q \in \Q$ and $\eta'
\in E_G$.
Since $\operatorname{char}(K)=0$ and $G$ is connected, $G(K)$ is divisible,
and the $n$-torsion is finite for all $n$ and hence contained in $G(C_0)$.
So $V := G(K)/G(C_0)$ is naturally a left $E^0_G$-module.
If $F \leq E^0_G$ is a division subring, we view $V$ as an $F$-vector space
and let $\P_F(G) := \P(V)$ be its projectivisation, and let $\eta_F^G :
\P_F(G) \rightarrow \G_K$ be the map induced by $g \mapsto \widetilde {g}$ for
$g \in G(K)$.
Note that $\eta_F^G$ is not injective.
\begin{example}
Let $G$ and $F$ be as above.
Let $g_i \in G(K)$ be independent generics over $C_0$ for $i$ in a (possibly
infinite) index set $I$.
Let $V := \left<{(g_i/G(C_0))_{i \in I}}\right>_F \leq G(K)/G(C_0)$.
Then $\eta_F^G$ maps the $|I|$-dimensional projective geometry
$\mathbb{P}_F(V) \subset \P_F(G)$ injectively into $\G_K$, and the image
$\eta_F^G(\mathbb{P}_F(V))$ is $\dim(G)$-dimensionally fully embedded in
$\G_K$.
For example, in the case $G=\mathbb{G}_m$, if $a_0,\ldots ,a_n \in K$ with
$\trd(\a/C_0)=n+1$, then they generate in $K^*/C_0^*$ the $\Q$-subspace $\{
\a^{\q} / C_0^* : \q \in \Q^{n+1} \} = \{ \prod_i a_i^{q_i} / C_0^* :
q_0,\ldots ,q_n \in \Q \}$; the algebraic dependencies over $C_0$ within
this set are precisely those arising from $\Q$-linear dependencies on the
exponents, and so this yields an embedding of $\P^n(\Q)$ in $\G_K$.
\end{example}
The following proposition, which is the main result of this appendix, says
that any fully embedded projective geometry (of sufficiently large dimension)
is of this form.
\begin{proposition} \label{prop:EHeq}
Let $(P,\cl) \subset \G_K$ be a $k$-dimensionally fully embedded geometry,
and suppose $P$ is isomorphic to the geometry of a projective space over a
division ring $F$, and $\dim(P) \geq 3$.
Then there is an abelian algebraic group $G$ over $C_0$ of dimension $k$,
and an embedding of $F$ as a subring $F \leq E^0_G$,
and a closed subgeometry $P'$ of $\P_F(G)$ on which $\eta_F^G$ is injective,
such that $P = \eta_F^G(P')$.
% such that the inclusion $P \subset \G_K$ factors via $\eta_F^G$;
% that is, $P = \eta_F^G(P')$ for some closed subgeometry $P'$ of $\P_F(G)$
% on which $\eta_F^G$ is injective.
% if $x_1,\ldots ,x_n \in P$ are independent,
% then there exist $g_1,\ldots ,g_n \in G$ such that $x_i = \widetilde {g_i}$
% and $\cl(x_1\ldots x_n) = \{ \Sigma \alpha_ig_i \modsim : \alpha_i \in \O
% \}$,
% where $\O := F \cap I'$.
Furthermore, $G$ is unique up to isogeny.
\end{proposition}
The remainder of this appendix constitutes a proof of
Proposition~\ref{prop:EHeq}.
The strategy of the proof is to find the commutative algebraic group $G$ via
the abelian group configuration theorem, and then to exhibit a natural
injective collineation from $P$ to $\P_F(G)$. The fundamental theorem of
projective geometry, in its version over division rings, then allows to claim
that this collineation is a projective embedding. However, since we must also
identify $F$ within $E^0_G$, we will in fact use a more general form of the
fundamental theorem.
\begin{remark}
In fact the proof applies directly to $C_0 \prec K$ models of an arbitrary
theory of finite Morley rank, with definable groups and endomorphisms in
place of algebraic groups and endomorphisms, as long as any connected
definable abelian group is divisible (equivalently, has finite $n$-torsion
for all $n$).
% finite Morley rank rather than just \omega-stable to ensure that
% invertible endomorphisms have finite kernel.
\end{remark}
\begin{remark}
Unlike in \cite{EH-projACF}, our techniques do not directly apply in the
case of $P$ a projective plane (i.e.\ a 3-dimensional connected modular
geometry), and so do not rule out non-Desarguesian projective planes.
However, David Evans has pointed out to us that the arguments of
\cite{Lindstrom-desarguesian} go through to show that any projective plane
appearing as a subgeometry of $\cl$ is Desarguesian (and hence is a
projective plane over a division ring).
\end{remark}
\begin{lemma} \label{lem:quadrCollapse}
Suppose $G$ is an abelian algebraic group over $C_0$,
and let $g,h \in G(K)$ and $b \in P$.
Suppose $\widetilde {g},\widetilde {h},b \in P$ are independent,
and $\widetilde {g+h} \in P$,
and $d \in \cl(\widetilde {g},b) \setminus \{\widetilde {g},b\}$.
Then there is $h' \in G(K)$ such that $\widetilde {h'} = b$ and $\widetilde
{g+h'}=d$.
\end{lemma}
\begin{proof}
By modularity, say
$c \in \cl(\widetilde {h},b) \cap \cl(\widetilde {g+h},d)$.
Then we proceed as in \cite[Lemma~1.2]{EH-autGeomACF}.
Say $b = \widetilde b'$ and $d = \widetilde d'$.
Then by the coheir property of independence in the stable theory $ACF$
any formula in $\tp(c/C_0b'd'g)$ is satisfiable in $C_0$.
So since $b'$ resp. $d'$ is interalgebraic over $c$ with points $x$ resp.
$y$ of $G$ satisfying $x+g=y$, they are already interalgebraic over $C_0$
with such points, as required.
(Alternatively, this argument can be phrased purely algebraically, by taking
a specialisation of $c$ to $C_0$ fixing $b'd'g$;
see \cite[Lemma~2.1.1]{EH-projACF}.)
\end{proof}
\begin{lemma} \label{lem:lines}
Suppose $a,b \in P$, $a\neq b$.
Then there is an abelian algebraic group $G$ over $C_0$ with $\dim(G)=k$,
and there exist $g,h \in G(K)$
such that $a = \widetilde {g}$, and $b = \widetilde {h}$, and
$\cl(a,b) \setminus \{a,b\} = P \cap \{ \widetilde {g + h} :
g,h \in G(K), a = \widetilde {g}, b = \widetilde {h} \}$.
\end{lemma}
\begin{proof}
This proof closely follows the argument of \cite[Theorem~3.3.1]{EH-projACF}.
As there, this proof actually only needs $\dim(P) \geq 3$, but we will make
the argument slightly more concrete by using that $P$ is a projective
geometry over $F$.
Take $c \in P \setminus \cl(a,b)$, and identify $\cl(a,b,c)$ with $\P(F^3)$,
placing $a,b,c$ at $[0:1:0],[0:0:1],[1:0:0]$ respectively.
Consider the following configuration.
\[ \xymatrix{
& & &&&& [0:1:0]=a \ar@{-}'[ddll][dddlll] \ar@{-}'[dlll][ddllllll] \\
& & & [1:1:0] \ar@{-}'[d][dd] & & & \\
c=[1:0:0] & & & [2:1:1] & [1:1:1] \ar@{-}'[l][llll] & & \\
& & & [1:0:1] && & \\
&& & & & & [0:0:1]=b \ar@{-}'[uull][uuulll] \ar@{-}'[ulll][uullllll]
} \]
By the assumption that $P$ is fully embedded in $\G_K$, this satisfies the
assumptions of Fact~\ref{fact:abGrpConf}.
Hence there exist an abelian algebraic group $G$ over $C_0$ and generics
$g,k \in G(K)$ such that
\begin{align*}
a=[0:1:0] &= \widetilde {g}\\
c=[1:0:0] &= \widetilde {k}\\
[1:1:0] &= \widetilde {g+k} .
\end{align*}
Now let $d \in \cl(a,b) \setminus \{a,b\}$. Then by
Lemma~\ref{lem:quadrCollapse} applied to $g,k,b$, we have $d=\widetilde
{g+h}$ for some $h \in G(K)$ with $\widetilde {h}=b$. The converse inclusion
is clear.
\end{proof}
Now let $a_0,b_0 \in P$, $a_0 \neq b_0$, and fix $G$ and $g_0,h_0 \in G(K)$ as
provided by this lemma, with $\widetilde {g_0}=a_0$ and $\widetilde
{h_0}=b_0$.
Say $g \in G(K)$ is \defnstyle{$P$-aligned} if there exists $g' \in G(K)$ with
$\widetilde {g'} \neq \widetilde {g}$ such that $\widetilde {g},\widetilde
{g'},\widetilde {g+g'} \in P$. Such a $g'$ is a \defnstyle{$P$-alignment
witness} for $g$.
\begin{lemma} \label{lem:alignedEx}
Every point of $P$ is of the form $\widetilde {g}$ for some $P$-aligned $g
\in G(K)$.
\end{lemma}
\begin{proof}
Let $c \in P$.
If $c \in \cl(a_0,b_0)$, such a $g$ exists by Lemma~\ref{lem:lines}.
Else, it follows from Lemma~\ref{lem:quadrCollapse} applied to $g_0,h_0,c$.
\end{proof}
\begin{lemma} \label{lem:alignedWitness}
If $g \in G(K)$ is $P$-aligned and $b \in P \setminus \widetilde {g}$,
then there exists a $P$-alignment witness $g' \in G(K)$ for $g$ with
$\widetilde {g'} = b$.
\end{lemma}
\begin{proof}
By Lemma~\ref{lem:quadrCollapse},
if $g'' \in G(K)$ is a $P$-alignment witness for $g$ and
$b \notin \cl(\widetilde {g},\widetilde {g''})$, then there exists a
$P$-alignment witness $g'$ for $g$ with $\widetilde {g'} = b$.
To handle the case that $b \in \cl(\widetilde {g},\widetilde {g''})$, apply
this first to $b' \in P \notin \cl(\widetilde {g},\widetilde {g''})$ to
obtain a witness $g'$, and then again to $b \notin \cl(\widetilde
{g},\widetilde {g'})$.
\end{proof}
\begin{lemma} \label{lem:alignedUnique}
If $g,h \in G(K)$ are $P$-aligned and $\widetilde {g} = \widetilde {h}$,
then $E^0_G(g/G(C_0))=E^0_G(h/G(C_0))$.
\end{lemma}
\begin{proof}
Say $g',h' \in G(K)$ are $P$-alignment witnesses for $g,h$ respectively.
By Lemma~\ref{lem:alignedWitness}, we may assume $\widetilde {g'} \notin
\cl(\widetilde {h},\widetilde {h'})$.
By Lemma~\ref{lem:quadrCollapse}, there is $h'' \in G(K)$ such that
$\widetilde {h''} = \widetilde {h'}$ and $\widetilde {g+h''} = \widetilde
{h+h'}$.
Then by Fact~\ref{fact:corrIsog},
there is $n \in \N$ and an isogeny $\alpha \in E_G$ and $k \in G(C_0)$ such
that $\alpha g = n h + k$,
as required.
\end{proof}
\begin{lemma} \label{lem:alignedLines}
Suppose $g,h \in G(K)$ are $P$-aligned and $\widetilde {g} \neq \widetilde
{h}$.
Then $\cl(\widetilde {g},\widetilde {h}) = P \cap \{ \widetilde {k} :
k/G(C_0) \in E_G^0(g/G(C_0)) + E_G^0(h/G(C_0)) \}$.
\end{lemma}
\begin{proof}
By Lemma~\ref{lem:alignedWitness}, we may take a $P$-alignment witness $g''$
for $g$ with $\widetilde {h} \notin \cl(\widetilde {g},\widetilde {g''})$.
Let $d \in \cl(\widetilde {g},\widetilde {h}) \setminus \{\widetilde
{g},\widetilde {h}\}$.
Then by Lemma~\ref{lem:quadrCollapse},
$d = \widetilde {g+h'}$ for some $h' \in G(K)$ with $\widetilde {h'} =
\widetilde {h}$.
Then $\widetilde {h'}$ is $P$-aligned, so by Lemma~\ref{lem:alignedUnique},
$h'/G(C_0) \in E_G^0(h/G(C_0))$.
\end{proof}
Our aim is to recognise $F$ as a subring of $E^0_G$, and $P$ as embedded in
the corresponding $F$-projectivisation of a subspace of $G(K)/G(C)$.
This is a matter of the fundamental theorem of projective geometry.
However, since $E^0_G$ is not necessarily a field, this is not the classical
case of the fundamental theorem. We use instead a version for projective
spaces over rings obtained by Faure \cite{Faure-projGeomRings}.
The following definitions are adapted from \cite{Faure-projGeomRings}.
\begin{definition}
The \defnstyle{projectivisation} $\P(M)$ of a module $M$ over a ring $R$ is
the set of non-zero 1-generated submodules $Rx$ equipped with the closure
operator $\cl_{\P(M)}$ induced from $R$-linear span,
\[ \cl_{\P(M)}( (Ry_i)_i ) := \{ Rx : x \in \left<{(y_i)_i}\right>_R \} .\]
A point $B \in \P(M)$ is \defnstyle{free} if $B = Rx$ for some $x \in M$ for
which $\{ \lambda \in R : \lambda x = 0 \} = \{0\}$.
If $N$ is a module over a ring $S$, a map $g : \P(M) \rightarrow \P(N)$ is a
\defnstyle{projective morphism} if $g(\cl_\P(M)(A)) \subset \cl_\P(N)(g(A))$
for any $A \subset \P(M)$.
If $\sigma : R \rightarrow S$ is a homomorphism, an additive map $f : M
\rightarrow N$ is \defnstyle{$\sigma$-semilinear} if $f(\lambda x) =
\sigma(\lambda) f(x)$ for any $x \in M$ and $\lambda \in R$. If $f$ is
injective, $\P(f) : \P(M) \rightarrow \P(N)$ is the induced projective
morphism.
A ring $R$ is \defnstyle{directly finite} if $(\forall \lambda,\mu \in R)
(\lambda\mu=1 \Rightarrow \mu\lambda=1)$.
\end{definition}
\begin{fact} \label{fact:faure}
Suppose $M$ and $N$ are modules over rings $R$ and $S$ respectively,
and $S$ is directly finite.
Suppose $g : \P(M) \rightarrow \P(N)$ is a projective morphism,
and $\im(g)$ contains free points $B_1,B_2,B_3$
such that for any $C_1,C_2 \in \im(g)$, for some $i \in \{1,2,3\}$, we have
\begin{equation}
\tag{C3}
\cl_{\P(N)}(C_1,C_2) \cap \cl_{\P(N)}(B_i) = \emptyset .
\end{equation}
Then there exists an embedding $\sigma : R \rightarrow S$ and a
$\sigma$-semilinear embedding $f : M \rightarrow N$ such that $g = \P(f)$.
\end{fact}
\begin{proof}
This is the statement of \cite[Theorem~3.2]{Faure-projGeomRings} in the case
$E=\emptyset $,
except that there $C_1$ and $C_2$ are not restricted to $\im(g)$; however, in
the proof the condition is used only when $C_1$ and $C_2$ are in $\im(g)$.
% Note: we are basically using the full power of Faure's result, and his
% proof seems to be essentially optimal. So referring to him rather than
% trying to give our own proof seems the correct move.
\end{proof}
Let $\P(G) := \P_{E^0_G}(G)$ be the projectivisation of the $E^0_G$-module
$G(K)/G(C_0)$.
Then Lemma~\ref{lem:alignedEx} and Lemma~\ref{lem:alignedUnique}
establish a map $\iota : P
\ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow} \P_{E^0_G}(G)$, which by
Lemma~\ref{lem:alignedLines} is a projective morphism.
We proceed to verify the assumptions of Fact~\ref{fact:faure}.
$E^0_G$ is directly finite since if $\mu\lambda=1$ with $\mu,\lambda \in E_G$
then $\mu$ is an isogeny so has a quasi-inverse $\mu' \in E_G$ with $n :=
\mu'\mu \in \N_{>0}$;
then $\lambda\mu n = n\lambda\mu=\mu'\mu\lambda\mu=\mu'\mu=n$,
so $\lambda\mu=1$ since $G$ is $n$-divisible.
Now $\dim(P) \geq 3$, so say $\widetilde {g}_1,\widetilde {g}_2,\widetilde
{g}_3 \in P$ are independent with $g_i$ being $P$-aligned.
Let $B_i := \iota(a_i) = E^0_G g_i$. Since each $g_i$ is generic in $G(K)$,
each $B_i$ is free.
To check (C3), suppose $C_i = \iota(\widetilde {h}_i)$ for $i=1,2$ with $h_i$
being $P$-aligned.
Then some $\widetilde {g}_i \notin \cl(\widetilde {h}_1,\widetilde {h}_2)$,
and so since $P$ is fully embedded in $\G_K$, we have $g_i \ind^0 h_1h_2$,
from which (C3) follows.
Now say $V$ is an $F$-vector space such that $P \cong \P(V)$.
Then by Fact~\ref{fact:faure},
for some embedding $\sigma : F \rightarrow E^0_G$ and $\sigma$-semilinear
embedding $f : V \rightarrow G(K)/G(C_0)$, we have $\iota = \P(f)$.
The main statement of Proposition~\ref{prop:EHeq} follows.
For the uniqueness up to isogeny of $G$, suppose Proposition~\ref{prop:EHeq}
also holds for a group $G'$.
then if $g,h \in G(K)$ with $\widetilde {g},\widetilde {h} \in P$ and
$\widetilde {g} \neq \widetilde {h}$, then, as in the proof of
Lemma~\ref{lem:alignedLines}, there are $g',h' \in G'(K)$ with $\widetilde
{g'}=\widetilde {g}$ and $\widetilde {h'}=\widetilde {h}$ and $\widetilde
{g+h}=\widetilde {g'+h'}$.
So by Fact~\ref{fact:corrIsog}, $G'$ is isogenous to $G$.
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