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\title{Incidence bounds in positive characteristic via valuations and
distality}
\author{Martin Bays \& Jean-François Martin}
\begin{abstract}
We prove distality of quantifier-free relations on valued fields with finite
residue field. By a result of Chernikov-Galvin-Starchenko, this yields
Szemerédi-Trotter-like incidence bounds for function fields over finite
fields. We deduce a version of the Elekes-Szabó theorem for such fields.
\end{abstract}
\maketitle
\section{Introduction}
We obtain the following incidence bound.
\begin{theorem} \label{t:SzTBasic}
Let $p$ be a prime, and let $K$ be a finitely generated extension of
$\F_p$.
Let $E \subseteq K^n \times K^m$ be the zero set of a set of polynomials in
$K[x_1,\ldots ,x_{n+m}]$.
Let $d,s \in \N$ and suppose $E$ is $K_{d,s}$-free, i.e.\ if $A\times B \subseteq E$
then $|A| < d$ or $|B| < s$.
Then there exists $\epsilon > 0$ (which can in principle be calculated as a
function of the number and degrees of the polynomials defining $E$) and $C >
0$ such that for any finite subsets $A \subseteq K^n$ and $B \subseteq K^m$,
$$|E \cap (A \times B)| \leq C(|A|^{1-\epsilon}
|B|^{\frac{d-1}d(1+\epsilon)} + |A| + |B|).$$
\end{theorem}
\subsection{Background and motivation}
The Szemerédi-Trotter theorem bounds the number of point-line incidences
between a set $P$ of points and a set $L$ of lines in the real plane.
We state a version with an explicit bound, \cite[Theorem~8.3]{TaoVu}:
\begin{fact} \label{f:STOrig}
For any finite $P$ and $L$,
$$|\{ (p,l) \in P \times L : p \in l\}|
\leq 4|P|^{\frac23}|L|^{\frac23}+4|P|+|L|.$$
\end{fact}
Statements of the form of Theorem~\ref{t:SzTBasic} can be seen as generalisations of
this, replacing the point-line incidence relation with other algebraic binary
algebraic relations. Such results were proven for characteristic 0 fields
first in \cite[Theorem~9]{ES}, and subsequently strengthened in
\cite[Theorem~1.2]{FoxEtAl}. Using such bounds for binary relations,
Elekes-Szabó \cite{ES} obtained somewhat analogous bounds for ternary
algebraic relations.
In positive characteristic, versions of Fact~\ref{f:STOrig} have been proven
(\cite{BKT},\cite{SZ}) where one restricts to sets which are small compared to
the characteristic. This is related to the sum-product phenomenon in fields,
where finite fields are known to be the only obstruction (\cite{BKT},
\cite[Theorem~2.55]{TaoVu}).
Meanwhile, a special case of results in \cite{CGS} and \cite[Section~2]{CS}
yields a version of Theorem~\ref{t:SzTBasic} in characteristic 0 by seeing it as a
consequence of the fact that the complex field is a reduct of a distal
structure, namely the real field.
The notion of distality and these incidence theoretic implications are
summarised in Section \ref{s:CSbackground} below.
It would be surprising if the positive characteristic results mentioned in the
previous paragraph, which require an unbounded characteristic, could be seen
as instances of distality. We consider instead the orthogonal situation of a
function field over a finite field, and we prove Theorem~\ref{t:SzTBasic} by finding
sufficient distality to trigger the incidence bounds of \cite{CGS}. We obtain
this distality using elementary notions from the model theory of valued
fields, and in fact our results apply more generally to any valued field with
finite residue field. Note that it follows from \cite{KSW} that a positive
characteristic valued field with finite residue field is not NIP, and so is
not the reduct of a distal structure; this forces us to use a more local
notion of distality.
Our motivation for considering these fields is \cite[Section~5]{Hr-psfDims},
which suggests a unifying explanation for all the results on existence of
bounds described above: they are all incarnations of {\em modularity} in the
model-theoretic sense, and they are consistent with a Zilber dichotomy
statement of the form ``any failure of modularity arises from an infinite
pseudofinite field''. In other words, finite fields should be the cause of any
failure of the bounds. As a special case, this would suggest that for a field
$K$ of characteristic $p > 0$ which has finite \defn{algebraic part} $\K \cap
\F_p^\alg$, incidence bounds and Elekes-Szabó results should go through as in
characteristic 0.
We partially confirm this only in the special\footnote{See
Proposition~\ref{p:latentFields}} case of fields admitting finite residue field.
However, in Theorem~\ref{t:ES} we do confirm for such $K$ that an Elekes-Szabó
result applies: a mild strengthening of Theorem~\ref{t:SzTBasic} suffices as input
to the proof of one of the main results of \cite{BB-cohMod}, yielding
Elekes-Szabó bounds for arbitrary arity algebraic relations in $K^n$ which do
not arise from 1-dimensional algebraic groups.
\subsection{Acknowledgements}
Thanks to Artem Chernikov and Sergei Starchenko for conversation which
launched the project, to Sylvy Anscombe, Philipp Dittmann, Udi Hrushovski, and
Silvain Rideau-Kikuchi for miscellaneous helpful conversation, and to
Elisabeth Bouscaren for matchmaking and sanity checking.
\emph{\small{Bays was supported in part by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) under Germany's Excellence Strategy EXC
2044–390685587, Mathematics Münster: Dynamics–Geometry–Structure.}}
% TODO: thank ANR for funding Jean-François's visit to Münster.
\section{Preliminaries}
We use basic notions and notation from model theory.
Let $\Lang$ be a (possibly many-sorted) first order language and $T$ a a
complete $\Lang$-theory.
\begin{notation}
If $\M \vDash T$ and $B \subseteq \M$ and $x=(x_1,\ldots ,x_n)$ is a tuple of variables of
sorts $S_1,\ldots ,S_n$, we write $B^x$ for $\prod_i (S_i(\M) \cap B)$.
We write $|x|$ for the length $|x| = n$ of the tuple.
For a set $B$,
we write $B_0 \subseteq_{\fin} B$ to mean that $B_0$ is a finite subset of $B$.
For a formula $\phi$, we define $\phi^0 := \neg\phi$ and $\phi^1 := \phi$.
If $\phi(x;y)$ is a partitioned formula and $b \in \M^x$ and $A \subseteq \M$, we
set $\tp_\phi(b/A) := \{ \phi(x,c)^\eps : c \in A^y;\; \eps \in \{0,1\};\; \M
\vDash \phi(b,c)^\eps \}$. The partitioning will often be left implicit.
\end{notation}
\section{Distality}
\subsection{Distal cell decompositions}
\label{s:CSbackground}
We recall the following definition from \cite{CGS}:
\begin{definition}
Let $A$ and $B$ be sets.
A binary relation $E \subseteq A \times B$ {\em admits a distal cell decomposition}
with exponent $t \in \R$ if there exist $s \in \N$ and finitely many relations
$\Delta_i \subseteq A \times B^s$ and $C \in \R$ such that for every $B_0 \subseteq_{\fin} B$,
$A$ can be written as a (not necessarily disjoint) union of $\leq C|B_0|^t$
subsets of the form $\Delta_i(c)$ for $c \in B_0^s$, each of which {\em cuts} no
$E(b)$ for $b \in B_0$, i.e.\ $E(b) \subseteq \Delta_i(c)$ or $E(b) \cap \Delta_i(c)
= \emptyset $.
\end{definition}
It was proven in \cite{CGS} that relations admitting distal cell
decompositions enjoy certain incidence bounds. For our purposes, the following
version of this deduced in \cite[Theorem~2.6,2.7(2)]{CS-ES1d} is most
relevant.
A binary relation $E \subseteq A \times B$ is \defn{$K_{d,s}$-free} if it contains no
subset $A_0 \times B_0$ with $|A_0| = d$ and $|B_0| = s$.
\begin{fact} \label{f:distInc}
Let $E \subseteq A \times B$ be $K_{d,s}$-free and admit a distal cell
decomposition with exponent $t$. Then for $A_0 \subseteq_{\fin} A$ and $B_0 \subseteq_{\fin} B$,
$$|E \cap (A_0 \times B_0)| \leq O_E(|A_0|^{\frac{(t-1)d}{td-1}}
|B_0|^{\frac{td-t}{td-1}} + |A_0| + |B_0|).$$
\end{fact}
\subsection{Distal subsets}
\begin{definition}
Let $\M \vDash T$.
Let $\phi(x;y)$ be an $\Lang(\M)$ formula, and let $A,B \subseteq \M$ be subsets.
\begin{itemize}\item An $\Lang$-formula $\zeta_\phi(x;z)$ is a \defn{uniform strong honest
definition} (\defn{USHD}) for $\phi$ on $A$ over $B$ if
for any $a \in A$ and finite subset $B_0 \subseteq_{\fin} B$ with $|B_0| \geq 2$,
there is $d \in B_0^z$
such that $\tp(a/B_0) \ni \zeta_\phi(x,d) \vdash \tp_\phi(a/B_0)$.
\item We omit ``on $A$'' in the case $A = \M$.
\item We omit ``over $B$'' in the case $B=A$.
\item $A$ is \defn{distal in} $\M$ if every $\Lang(A)$-formula $\phi(x;y)$ has
a USHD on $A$.
\end{itemize}
\end{definition}
The notion of a strong honest definition comes from \cite{CS-extDefII}. We
work with USHDs rather than directly with distal cell decompositions in order
to be able to reduce to one variable (Lemma~\ref{l:distal1}), and because dealing
with a single formula is more convenient for many purposes. As the following
remark makes explicit, there is little difference between the two notions.
\begin{remark} \label{r:distDist}
An $\Lang$-formula $\phi(x;y)$ has a USHD on $A$ over $B$ if and only if the
binary relation $E := \phi(A;B) \subseteq A^x\times B^y$ admits a distal cell
decomposition where the $\Delta_i$ are themselves defined by
$\Lang$-formulas. The restriction $|B_0| \geq 2$ allows multiple $\Delta_i$ to
be coded as one formula, a trick we will use repeatedly; explicitly, if
$\delta_i(x,z_i)$ define $\Delta_i$, then
$$\zeta(x,z_1,\ldots ,z_s,w_1,\ldots ,w_s,w'_1,\ldots ,w'_s) :=
\bigwedge_i (\delta_i(x,z_i) \leftrightarrow w_i = w'_i)$$
is a USHD for $\phi$ on $A$ over $B$.
In particular, if $A \subseteq \M$ is distal in $\M$, then the trace on $A$ of any
$\Lang(A)$-formula $\phi(x,y)$ admits a distal cell decomposition.
\end{remark}
\subsection{Reductions}
\begin{lemma} \label{l:distal1}
A subset $A \subseteq \M$ is distal in $\M$ if and only if any $\Lang$-formula
$\phi(x;y)$ with $|x| = 1$ has a USHD on $A$.
\end{lemma}
\begin{proof}
First, it follows by an inductive argument from the 1-variable case that any
$\Lang$-formula has a USHD on $A$; we refer to the proof of
\cite[Proposition~1.9]{ACGZ} for this argument.
It remains to deduce that any $\Lang(A)$-formula has a USHD on $A$, but it
follows directly from the definition that if $\phi(x;y,z)$ has a USHD on $A$
and $a \in A^z$, then $\phi(x;y,a)$ has a USHD on $A$.
\end{proof}
% (proof written before reading ACGZ's, not really any different)
%Proof:
% Let $\phi(x,x';y)$ be an $\Lang$-formula and suppose inductively that any
% $\Lang$-formula in $x$ or in $x'$ has a USHD on $B$.
% So there are $\zeta(x,x',z)$ and $\xi(x',z')$ such that given $b \in B^x$ and
% $b' \in B^{x'}$ and a finite subset $B_0 \subseteq B$ with $|B_0|\geq 2$, there are $d
% \in B_0^z$ and $d' \in B_0^{z'}$ such that
% $$\tp(b/B_0b') \ni \zeta(x,b',d) \vdash \tp_{\phi(x;x',y)}(b/B_0b')$$
% (the existence of such a $\zeta$ follows straightforwardly from existence of
% a USHD for $\phi(x;x',y)$ and using coding to separate out $b'$),
% and, setting $$\theta(x',z,y) := \forall x. (\zeta(x,x',z) \rightarrow
% \phi(x,x',y)),$$
% we have $$\tp(b'/B_0) \ni \xi(y,d') \vdash \tp_{\theta}(b'/B_0) \supseteq \{
% \theta(y,d,a_0) : a_0 \in B_0^y, \vdash \phi(b,b',a_0) \}.$$
%
% But then $\tp(bb'/B_0) \ni (\zeta(x,x',d) \wedge \xi(x',d')) \vdash \{ \phi(x,x',a_0)
% : a_0 \in B_0^y, \vdash \phi(b,b',a_0) \}$.
%
% Repeating with $\neg\phi$, we conclude that $\phi(x,x';y)$ has a USHD on
% $B$.
% .
\begin{lemma} \label{l:dd-feq}
Let $\M$ be an $\Lang$-structure.
Let $S$ and $\widetilde {S}$ be $\Lang$-sorts and let $f : \widetilde {S} \rightarrow S$ be an
$\Lang$-definable function with uniformly finite fibres,
say $|f^{-1}(b)| \leq N$ for all $b \in f(S)$.
Suppose $B \subseteq f(S(\M))$, and let $\widetilde {B} := f^{-1}(B) \subseteq \widetilde {S}$.
Let $A \subseteq \M^x$ and
let $\phi(x,y)$ be an $\Lang$-formula
such that $\phi(x;f(z))$ has a USHD on $A$ over $\widetilde {B}$.
Then $\phi(x;y)$ has a USHD on $A$ over $B$.
\end{lemma}
\begin{proof}
Say $\zeta(x,w)$ is a USHD for $\phi(x;f(z))$ over $\widetilde {B}$.
Let $B_0 \subseteq_{\fin} B$ and $a \in A$. Then $\widetilde {B}_0 := f^{-1}(B_0)$ is a finite subset
of $\widetilde {B}$, so there is $\widetilde {d}$ such that $\tp(a/\widetilde {B}_0) \ni \zeta(x,\widetilde {d}) \vdash
\tp_{\phi(x;f(z))}(a/\widetilde {B}_0) \vdash \tp_{\phi(x;y)}(a/B_0)$.
\newcommand{\epsbar}{\overline{\eps}}
Let $d := f(\widetilde {d})$. Then $|f^{-1}(d)| \leq N^{|w|}$, and so there is $M <
N^{|w|}$ and $\epsbar_0 \in \{0,1\}^M$ and $\b_0 \in (B_0)^M$ such that
$\theta_{M,\epsbar_0}(w,d,\b_0)$ has the minimal number of realisations
amongst the formulas
$$\theta_{n,\epsbar}(w,d,\b) := (f(w) = d \wedge \forall x. (\zeta(x,w) \rightarrow
\bigwedge_{i=1}^n \phi(x,b_i)^{\eps_i}))$$
which hold of $\widetilde {d}$, with $n \in \N$ and $\epsbar \in \{0,1\}^n$ and $\b \in
(B_0)^n$. The bound $M < N^{|w|}$ follows from the observation that if such
a formula does not have the minimal number of realisations, then a single
new instance of $\phi$ can be added to reduce the number of realisations.
By the minimality, we have for any $b \in B_0$ that
$\theta_{M,\epsbar_0}(w,d,\b_0) \vdash \forall x. (\zeta(x,w) \rightarrow
\phi(x,b)^\eps)$ for some $\eps \in \{0,1\}$.
So $\tp(a/B_0) \ni \exists w. (\theta_{M,\epsbar_0}(w,d,\b_0) \wedge \zeta(x,w))
\vdash \tp_{\phi(x;y)}(a/B_0)$. Coding the finitely many such formulas with $M <
N^{|w|}$ and $\epsbar_0 \in \{0,1\}^M$ into a single formula,
we therefore obtain a USHD for $\phi(x;y)$ on $A$ over $B$.
% The exponent of the distal cell decomposition does not increase.
% We can take a single compressing instance downstairs which will work for
% everything which a given \widetilde {d} works for. So there's no change at all in the
% number of cells.
\end{proof}
\begin{remark}
The finiteness assumption in Lemma~\ref{l:dd-feq} is necessary.
Consider for example the structure $(X,O_X;<)$ where $X$ is a set, $O_X$ is
the set of linear orders on $X$, and $x<_ox'$ is the corresponding ternary
relation. Let $\pi_1 : X \times O_X \rightarrow X$ be the projection.
As one may see by considering automorphisms, the induced structure on $X$ is
trivial, so $x = y$ has no USHD on $X$ over $X$.
But $x = \pi_1(z)$ has a USHD on $X$ over $X\times O_X$ (since if $X_0 \subseteq_{\fin}
X$ and $o \in O_X$, then $\tp_=(x/X_0)$ is implied by the $<_o$-cut of $x$ in
$X_0$).
\end{remark}
%Remark:
% The following nonstandard form of the above proof is neater but less
% explicit. Compressibility can be characterised as isolation by a
% non-standard parameter, so taking an $|\M|^+$-saturated $L_P$-elementary
% extension $(\M^*;B^*) \succeq (\M;B)$ and setting $\widetilde {B}^* := f^{-1}(B^*)$,
% we have $\tp(a/\widetilde {B}^*) \ni \zeta(x,\widetilde {b}^*) \vdash \tp_{\phi(x,f(z))}(a/\widetilde {B}) \vdash
% \tp_{\phi(x,y)}(a/B)$. Now $\widetilde {b}^* \in \acl(B^*)$, so say $\xi(w,b^*)$ isolates
% $\tp(\widetilde {b}^*/B^*)$. Then $\tp(a/B^*) \ni \exists w. (\xi(w,b^*) \wedge \zeta(x,w))
% \vdash \tp_{\phi(x,y)}(a/B)$.
% .
\subsection{Remarks}
We add some further remarks concerning these definitions, which will not be
used subsequently.
\begin{remark} \label{r:distalElem}
Suppose $B$ is distal in an $\Lang$-structure $\M$.
Then this is expressed in the $\Lang_P$-theory of $(\M;B)$, where $P$ is a
new predicate interpreted as $B$;
i.e.\ if $(\M';B') \equiv (\M;B)$,
then $B'$ is distal in $\M'$.
\end{remark}
\begin{remark}
By \cite[Theorem~21]{CS-extDefII}, $\Th(\M)$ is distal if and only if $\M$
is distal in $\M$.
(No saturation assumption is needed here, thanks to
Remark~\ref{r:distalElem}.)
\end{remark}
\begin{remark}
Distality in $\M$ of a subset $B \subseteq \M$ is equivalent to distality of the
induced structure $(B;(\phi(B)_{\phi \text{ an $\L$-formula}})$ if this
structure admits quantifier elimination, but in general is much weaker. We
could say that distality of a subset means that it has ``quantifier-free
distal induced structure''.
\end{remark}
\begin{example} \label{e:indSeq}
If $B = (b_i)_i \subseteq \M$ is an $\emptyset $-indiscernible sequence which is not
totally indiscernible, and this is witnessed by an $\Lang$-formula
$\theta_<$ with $\M \vDash \theta_<(b_i,b_j) \Leftrightarrow i0$, then the induced valuation
on the algebraic part $K\cap \F_p^\alg$ is trivial. So a positive
characteristic field which admits a valuation with finite residue field has
finite algebraic part.
However, the converse fails.
\begin{proposition} \label{p:latentFields}
For any prime $p$,
there exists an algebraic extension $L \geq \F_p(t)$ such that $L\cap
\F_p^\alg = \F_p$ but no valuation on $L$ has finite residue field.
\end{proposition}
\begin{proof}
We work in an algebraic closure $\F_p(t)^\alg$ of $\F_p(t)$.
Let $\wp : \F_p(t)^{\alg} \rightarrow \F_p(t)^\alg$ be the Artin-Schreier map
$\wp(x) := x^p-x$, an additive homomorphism with kernel $\F_p$.
\begin{claim*}
$\deg(\F_p(t,(\wp^{-1}(t^a))_{a > 0}) / \F_p(t))$ is infinite.
\end{claim*}
\begin{proof}
By \cite[Theorem~8.3]{Lang-algebra},
it suffices to see that $\{ t^a | a > 0\}$ is not contained in any finite
union of additive cosets of $\wp(\F_p(t))$.
Let $a,b \in \N \setminus p\N$ be distinct. Let $\beta_{a,b} := \sum_{i \geq 0}
(t^{ap^i} - t^{bp^i}) \in \F_p[[t]]$. Then $\wp(\beta_{a,b}) = t^b - t^a$.
Now $\beta_{a,b} \notin \F_p(t)$, since there are arbitrarily long intervals
between exponents with non-zero coefficient in this power series.
So $(t^a)_{a \in \N \setminus p\N}$ lie in distinct cosets of $\wp(\F_p(t))$.
\end{proof}
We write $\res$ for the residue field map associated to a chosen valuation
$v$ on a field $K$, and $\res(K)$ for the corresponding residue field.
\begin{claim*}
Let $K' \geq K \geq \F_p(t)$ be finite field extensions, and suppose $K' \cap
\F_p^\alg = \F_p$.
Let $v$ be a valuation on $K$ with $\res(K)$ finite.
Then there exists a finite field extension $K'' \geq K'$ such that
$K'' \cap \F_p^\alg = \F_p$
but for any extension of $v$ to $K''$, $\res(K'') \gneq \res(K)$.
\end{claim*}
\begin{proof}
The valuation $v$ is non-trivial, so say $v(s) > 0$.
So $v$ induces the $s$-adic valuation on $\F_p(s) \leq K$.
Now $s$ is transcendental, so $t$ is algebraic over $s$, so $K$ is also a
finite extension of $\F_p(s)$.
So we may assume without loss that $v$ restricts to the $t$-adic valuation
on $\F_p(t)$.
Since $\res(K)$ is finite, it is not Artin-Schreier closed;
say $\alpha \in \res(K) \setminus \wp(\res(K))$.
Say $\res(\bar\alpha) = \alpha$.
Since $\deg(K'/\F_p(t))$ is finite, it follows from the above Claim that
$$\deg(K'(\wp^{-1}(\bar\alpha),(\wp^{-1}(\bar\alpha + t^a))_{a > 0})/K')$$
is infinite.
So say $a>0$ is such that $K'' := K(\wp^{-1}(\bar\alpha+t^a)) \not\subseteq
K'(\F_{p^p})$.
Then by considering degrees, $K'' \cap \F_p^\alg = \F_p$.
But for any extension of $v$ to $K''$,
$$\wp(\res(\wp^{-1}(\bar\alpha+t^a))) = \res(\bar\alpha+t^a) = \alpha \notin
\wp(\res(K)),$$
so $\res(K'') \gneq \res(K)$.
\end{proof}
Now we recursively construct a chain $K_0 := \F_p(t) \leq K_1 \leq \ldots $
of finite extensions of $\F_p(t)$.
Let $\eta : \omega \times \omega \rightarrow \omega$ be a bijection such that
$\eta(i,j) \geq i$ for all $i,j$.
Note that $\F_p(t)$ admits only countably many valuations (identifying a
valuation with its valuation ring); indeed, as above, each non-trivial
valuation is a finite extension of the $s$-adic valuation on some $\F_p(s)
\leq \F_p(t)$; there are only countably many choices for $s$, and only
finitely many ways to extend a valuation to a finite extension
(\cite[Theorem~3.2.9]{EnglerPrestel}).
Hence also there are also only countably many valuations on each $K_i$.
Once $K_i$ is constructed, let $\{v_{i,j} : j \in \omega\}$ be the set of
valuations on $K_i$ with finite residue field.
Suppose $k=\eta(i,j)$ and $K_k$ has been constructed. Let $K_{k+1} \geq K_k$
be an extension as in the second Claim for the extensions $K_k \geq K_i \geq
\F_p(t)$ and the valuation $v_{i,j}$ on $K_i$.
Now let $K_\omega := \bigcup_{k < \omega} K_k$.
We have $K_\omega \cap \F_p^\alg = \F_p$ since this holds for each $K_k$.
Suppose $v$ is a valuation on $K_\omega$ with finite residue field.
Then $\res(K_\omega) = \res(K_i)$ say, and the restriction of $v$ to $K_i$
is $v_{i,j}$ say.
Then $\res(K_{\eta(i,j)+1}) = \res(K_i)$, contradicting the construction.
\end{proof}
\begin{remark}
One might expect that a Zorn argument could replace the recursive
construction of the previous Proposition, i.e.\ that any maximal regular
extension of $\F_p(t)$ has no valuation with finite residue field. But
$\F_p((t^\Q)) \cap \F_p(t)^\alg$ is a counterexample. Thanks to Zoé
Chatzidakis for pointing this out.
\end{remark}
\section{Distality in $\ACVF$ of subfields with finite residue field}
\subsection{Uniform Swiss cheese decompositions}
Let $L$ be a non-trivially valued algebraically closed field.
Write $v$ for the valuation map and $\res$ for the residue field map.
We consider $L$ as an $\Ldiv := \{+,-,\cdot,|,0,1\}$-structure, where $x|y \Leftrightarrow
v(x) \leq v(y)$; by a result of Robinson, $L$ has quantifier elimination in this
language.
An open resp.\ closed \defn{ball} in $L$ is a definable set of the form $\{ x :
v(x-a) > \alpha \}$ resp.\ $\{ x : v(x-a) \geq \alpha \}$, where $a \in L$ and
$\alpha \in v(L) \cup \{-\infty,+\infty\}$.
\begin{fact}[Canonical Swiss cheese decomposition] \label{f:swiss}
Any boolean combination of balls can be represented as a finite disjoint
union of ``Swiss cheeses'' $\bigcupdot_{i 1 \}$. If
$\eps = v(\gamma_1-\gamma_i)$ for some $i>1$, we conclude by the inductive
hypothesis. Otherwise, by the ultrametric triangle inequality,
$v(x-\gamma_i) = \eps$ if $\eps < v(\gamma_1-\gamma_i)$, and
$v(x-\gamma_i) = v(\gamma_1-\gamma_i)$ otherwise, so the affine constraint
is equivalent to $\eps < \nu'$ or $\eps > \nu'$ for some $\nu'$.
\end{proof}
We may assume $L$ is $\aleph_0$-saturated, and so by compactness we obtain a
bound on the number of balls involved in this boolean combination which is
uniform in $a$.
So $\phi(x,a)$ is a boolean combination of boundedly many balls each having
a point in an extension of the subfield generated by $a$ of degree dividing
$$d := \lcm_i (\lcm(\deg_x f_i, \deg_x g_i)),$$
and it follows that the rounds and holes in the Swiss cheese decomposition
also have this property and are bounded in number.
\end{proof}
\subsection{Compressing cheeses}
Let $B$ be the imaginary sort of $L$ consisting of balls, both open and
closed, including the empty ball and its complement. We write $x \in b$ for the
corresponding $\emptyset $-definable (in $\Leq$) element relation $( \in ) \subseteq L \times
B$.
Given $N \in \N$, let $S_N$ be the imaginary sort of $L$ which codes Swiss
cheese decompositions of complexity at most $N$. This means that we have an
associated $\emptyset $-definable element relation, which we also write as $( \in ) \subseteq
L \times S_N$, such that $c_1 = c_2$ iff $\{ x : x \in c_1 \} = \{ x : x \in c_2
\}$, and
setting $$X_N := \{(b_i)_{i 1$.
Say $p_i \in b_i$ is of degree dividing $d$ over $K$, and let $\alpha \in
v(L)$ be the valuative radius of $b$.
Then $v(p_i-p_j)=\alpha$ for $i\neq j$, since $b_i \vee b_j = b$ (in particular,
$b \neq L$).
Then $i \mapsto \lambda_i := \res(\frac{p_i-p_1}{p_2-p_1})$ is an injection of
$\{1,\ldots ,s\}$ into $\res(L)$.
Indeed, if $\lambda_i=\lambda_j$ then $\res(\frac{p_i-p_j}{p_2-p_1}) = 0$,
so $v(p_i-p_j) > v(p_2-p_1) = \alpha$, so $i=j$.
Since each $\lambda_i$ is in the residue field of an extension of $K$ of
degree dividing $d^3$,
we have $\lambda_i \in \F_{q^{d^3}}$ by the valuation inequality
(\cite[Corollary~3.2.3]{EnglerPrestel}).
So $s \leq q^{d^3}$.
% So we get an exponent 2(q^{d^3}+1)
\item
$x \in f_{S_N}(y)$ is equivalent, by the definition of $f_{S_N}$, to a
certain boolean combination of the formulas $(x \in y_i)_{i < N(N+1)}$.
So by (i), coding these formulas yields a formula which is a USHD for $x \in
f_{S_N}(y)$ over $B_{K,d}$.
% The exponent doesn't change (we just get more \Delta_i, one for each
% co-ordinate).
\item
Considering now $X_N$ as a sort and $y$ as a variable of sort $X_N$,
it follows from (ii) that $x \in f_{S_N}(y)$ has a USHD over $X_N(B_{K,d})$.
Then we conclude by Lemma~\ref{l:dd-feq}.
\end{enumerate}
\end{proof}
\subsection{Concluding distality}
\begin{lemma} \label{l:fin-res-comp}
Let $L$ be a non-trivially valued algebraically closed field.
Let $K \leq L$ be a subfield and suppose $\res(K)$ is finite.
Let $\phi(x;y)$ be an $\Ldiv$-formula with $|x| = 1$.
Then $\phi$ has a USHD over $K$.
Moreover, for any $r \geq 1$, $\phi$ has a USHD over the set $K_r \subseteq K^{\alg}
\subseteq L$ of points with degree over $K$ dividing $r$,
$$K_r := \{ a \in L : \deg(K(a)/K) | r \}$$
\end{lemma}
\begin{proof}
Let $N$ and $d$ be as in Lemma~\ref{l:ACVFQE-uniformity} for $\phi$.
Then there is a $\emptyset $-definable function $h : L^{|y|} \rightarrow S_N$ such that
$L^{\eq} \vDash \forall x,y. (\phi(x,y) \leftrightarrow x \in h(y))$,
and $h(K_r) \subseteq S_N(B_{K,dr})$.
By Lemma~\ref{l:B-rel-distal-local}(iii),
say $\zeta(x,z'_1,\ldots ,z'_s)$ is a USHD for $x \in y$ over $S_N(B_{K,dr})$.
Then (an $\Ldiv$-formula equivalent to) $\zeta(x,h(z_1),\ldots ,h(z_s))$ is a
USHD for $\phi(x;y)$ over $K_r$.
\end{proof}
\begin{theorem} \label{t:fin-res-distal}
Let $K$ be a valued field with finite residue field.
Let $L \geq K$ be an algebraically closed valued field extension.
Then $K$ is distal in $L$,
as is each $K_r$ defined as in Lemma~\ref{l:fin-res-comp}.
\end{theorem}
\begin{proof}
We may assume that $L$ is non-trivially valued, as otherwise $K$ is finite
and the result is trivial.
The result then follows from Lemma~\ref{l:fin-res-comp} and Lemma~\ref{l:distal1}.
\end{proof}
\begin{remark}
This does not reprove distality of $\Q_p$, because $\Q_p$ does not eliminate
quantifiers in $\Ldiv$.
\end{remark}
\section{Incidence theory consequences}
\begin{theorem} \label{t:SzT}
Let $K$ be a valued field with finite residue field.
Let $E \subseteq K^n \times K^m$ be quantifier-free definable in $\Ldiv(K)$.
Suppose $E$ omits $K_{d,s}$, where $d,s \in \N$.
Then there exist $t$ (see Remark~\ref{r:bounds}) and $C>0$ such that
for $A_0 \subseteq_{\fin} K^n$ and $B_0 \subseteq_{\fin} K^m$,
$$|E \cap (A_0 \times B_0)| \leq C(|A_0|^{\frac{(t-1)d}{td-1}}
|B_0|^{\frac{td-t}{td-1}} + |A_0| + |B_0|).$$
The same holds if $K$ is replaced by $K_r \subseteq K^{\alg}$ defined as in
Lemma~\ref{l:fin-res-comp}.
\end{theorem}
\begin{proof}
By Theorem~\ref{t:fin-res-distal} and Remark~\ref{r:distDist},
$E$ admits a distal cell decomposition, and we conclude by Fact~\ref{f:distInc}.
\end{proof}
The version of this stated in the introduction, Theorem~\ref{t:SzTBasic}, follows by
considering Lemma~\ref{l:fgRes} and the special case that $E$ is defined as the zero
set of polynomials over $K$, and setting $\epsilon := \frac1{dt-1}$.
\begin{remark} \label{r:bounds}
By examining the proof, in the case $n=1$ one can obtain a bound on the
exponent of the resulting distal cell decomposition giving $t \leq
2(q^{d^3}+1)$ where $q = |\res(K)|^r$ and $d$ is as in the proof of
Lemma~\ref{l:fin-res-comp}. Indeed, this is the exponent arising from bounding the
number of balls used in Lemma~\ref{l:B-rel-distal-local}(i), and neither
Lemma~\ref{l:B-rel-distal-local}(ii) nor Lemma~\ref{l:dd-feq} increase the exponent (for
the latter case, this follows from the structure of the proof, since each
instance $\zeta(x,\widetilde {d})$ gives rise to a single instance of the eventual
formula).
So we obtain the corresponding explicit bounds in Fact~\ref{f:distInc}. However,
we have no reason to expect these bounds to be anything like optimal.
For $n>1$, calculating explicit bounds is complicated by the fact that when
reducing to one variable a USHD for a quantified formula is used, so one
needs a bound on the degrees in the quantifier-free formula obtained by
quantifier elimination in ACVF. This quantifier elimination is primitive
recursive \cite{Weispfenning}, so in principle this could be done, yielding
an effective algorithm for computing an exponent $t$ for a given $E$
(uniform in definable families). But we do not attempt to make this explicit
here.
% Note: https://arxiv.org/abs/1511.08123 might be relevant
Instead, we illustrate the idea by showing that in the special case of
Szemerédi-Trotter, $E = \{ ((x,y),(a,b)) : y=ax+b \}$, we can take $t :=
4(q+1)$.
\newcommand{\y}{{\overline{y}}}
\newcommand{\z}{{\overline{z}}}
\newcommand{\w}{{\overline{w}}}
The proof of Lemma~\ref{l:B-rel-distal-local}(i) in this case gives a USHD
$\zeta(y,x,\z)$ for $\phi(y;x,(a,b)) := (x,y)E(a,b)$ over $K$, expressing
that $y$ is an element of a boolean combination of the points
$z_{i,1}x+z_{i,2}$ and the balls spanned by pairs of such points, with at
most $2(q+1)$ such points involved. Using coding to choose the form of the
boolean combination, this has exponent $\leq 2(q+1)$.
By \cite[Theorem~2.1]{Weispfenning}, if an $\Ldiv$ qf-formula
$\psi(x,\y,\z)$ is linear in $x,\y$, i.e.\ each polynomial has degree 1 in
$x$ and each $y_i$, then $\exists x. \psi(x,\y,\z)$ is equivalent modulo
$\ACVF$ to a qf-formula linear in $\y$.
% Note there seems to be a typo in the proof of 2.3 in Weispfenning, the
% equality in $\psi_2$ should be ">".
Now the formula $\zeta(y,x,\z) \rightarrow (x,y)Ew$ is linear in $x,y$,
so $\forall y. (\zeta(y,x,\z) \rightarrow (x,y)Ew)$ is equivalent to a qf-formula
which is linear in $x$.
Similarly $\forall y. (\zeta(y,x,\z) \rightarrow \neg(x,y)Ew)$ is equivalent to a
qf-formula linear in $x$, and the two can be coded into a single qf-formula
linear in $x$. This then itself admits (by the $n=1$ case of the present
Remark with $d=1$) a USHD $\xi(x,\w)$ over $K$ of exponent $\leq 2(q+1)$.
Then $\xi(x,\w) \wedge \zeta(y,x,\z)$ is a USHD for $E$ over $K$ of exponent
$\leq 2(q+1) + 2(q+1) = 4(q+1)$.
% Note: should be straightforward to make this work for arbitrary linear $E$,
% giving $2n(q+1)$, but it seems awkward to write.
In symmetric form, this gives a bound of $O(N^{\frac32 -
\frac1{16(q+1)-2}})$ on the number of incidences of $N$ lines and $N$
points.
As a final remark, note that although Theorem~\ref{t:SzT} does apply in
characteristic 0, e.g.\ to $K=\Q_p$, the bounds we obtain in this way are
worse than those obtained from \cite{FoxEtAl} by embedding $\Q_p$ in
$\C=\R^2$, even for $p=2$.
\end{remark}
\begin{question}
Is the dependence on $q$ in these bounds necessary?
For example, does there exist $\epsilon>0$ such that for all primes $p$
there exists $C$ such that for all $X,A \subseteq \F_p(t)^2$ we have
$|\{((x,y),(a,b)) \in X\times A : y=ax+b\}| \leq
C\max(|X|,|A|)^{\frac32-\epsilon}$?
(Remark~\ref{r:bounds} yields a bound depending on $p$ of $\epsilon =
\frac1{16p+14}$ in this case. Meanwhile one can obtain a lower bound
exponent of $\frac43$ by considering a rectangular example with bounded
degree polynomials, $\F_p[t]_{ 0$ such that
for all $X_i \subseteq_{\fin} K_0$, $i=1,\ldots ,n$, we have
$$|V(K_0) \cap \prod_i X_i| \leq C(\max_i |X_i|)^{d-\eps}.$$
\item $V$ is \emph{special}: $V$ is in co-ordinatewise
correspondence\footnote{As defined in \cite[Definition~1.1]{BB-cohMod}} with
a product $\prod_i H_i \leq \prod G_i^{n_i}$ of connected subgroups of $H_i$
of powers $G_i$ of 1-dimensional algebraic groups.
\end{enumerate}
\end{theorem}
\begin{proof}
\renewcommand{\a}{{\overline{a}}}
Let $K' \leq K$ be as above. Also let $C_0 \leq K'$ be a countable algebraically
closed subfield over which $V$ is defined.
The proof in \cite{BB-cohMod} goes through, but using Theorem~\ref{t:SzT} in place
of \cite[Theorem~2.14]{BB-cohMod}, and with
\cite[Theorem~3.3.1]{EvansHrushovski} replacing \cite[Proposition
A.4]{BB-cohMod}. We describe the necessary changes.
Firstly, \cite[Theorem~2.15]{BB-cohMod} goes through in the case that $X_i
\subseteq ((K_r)^\U)^{n_i}$ for some $r$ ($i=1,2$). The proof is identical, using
Theorem~\ref{t:SzT}; the sublinearity of the dependence on $s$ where $K_{2,s}$ is
omitted, discussed after \cite[Theorem~2.14]{BB-cohMod}, also holds here:
this is described in \cite[Remark~2.7(2),Corollary~2.8]{CS-ES1d}, and is
proven explicitly in \cite[Theorem~2.6]{CPS-ES}. (In fact this sublinearity
isn't necessary for the present 1-dimensional case.)
Now \cite[Theorem~5.9]{BB-cohMod} goes through for $P \subseteq (K')^{<\omega}$.
The proof is identical, except that in the proof of
\cite[Proposition~5.14]{BB-cohMod}, since $\a,\d \in (K')^{<\omega}$, already
$\a,\d \in ((K_r)^\U)^{<\omega}$ for some $r$, and this passes through to the
types $X_i$ since $(K_r)^\U$ is definable, so the above restricted form of
\cite[Theorem~2.15]{BB-cohMod} applies.
Next, $\End^0_{C_0}(G)$ must be redefined as the skew-field of quotients of
$\End_{C_0}(G)$ (this agrees with $\Q\otimes\End_{C_0}(G)$ in characteristic
0); see \cite[3.1]{EvansHrushovski} for discussion of the possibilities.
\newcommand{\G}{{\mathcal{G}}}
\newcommand{\x}{{\overline{x}}}
\newcommand{\h}{{\overline{h}}}
Finally, we indicate how to circumvent the use of \cite[Proposition
A.4]{BB-cohMod}, which is proven only in characteristic 0, in the
1-dimensional case. Where this is applied in
\cite[Proposition~6.1]{BB-cohMod},
we have $a_i \in K'$ ($i=1,\ldots ,n$) such that $\G_\a = \{ \acl^0(a_i) : i \}$
embeds in a projective subgeometry of the $\acl^0$-geometry $\G_K$ of $K$.
(Here we have $a_i \in K'$ rather than $a_i \in K^{<\omega}$, as this is what
arises in the proof, via \cite[Theorem~7.4]{BB-cohMod}, in the 1-dimensional
case corresponding to the statement of the current theorem.)
By \cite[Theorem~3.3.1]{EvansHrushovski}, there is a 1-dimensional algebraic
group $G$ over $C_0$ and generic $\x \in G^m$ over $C_0$ (where
$m=\dim(\G_\a)$) and $A \in \operatorname{Mat}_{n,m}(\End(G))$ such that, setting $\h :=
A\x$, we have $\acl^0(h_i) = \acl^0(a_i)$. Then $\operatorname{loc}^0(\h) = AG^m$ is a
connected algebraic subgroup of $G^n$, as required.
The rest of the proof goes through unchanged.
\end{proof}
\begin{remark}
The only obstruction to pushing this to higher dimension, i.e.\ to a version
of \cite[Theorem~1.11]{BB-cohMod}, is the need to generalise the higher
dimensional version of Evans-Hrushovski \cite[Proposition A.4]{BB-cohMod} to
positive characteristic.
Meanwhile, the proof of the converse direction (showing that every special
variety admits no powersaving) makes essential use of the characteristic 0
assumption in \cite[Proposition~7.10]{BB-cohMod}; this may not be so easy to
generalise, and the statement may need to change.
For these reasons, we leave positive characteristic analogues of
\cite[Theorem~1.11]{BB-cohMod} to future work.
\end{remark}
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