# Incidence bounds in positive characteristic via valuations and distality

Martin Bays & Jean-François Martin

###### 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.

##  1\. Introduction

We obtain the following incidence bound.

######  Theorem 1.1.

Let $p$ be a prime, and let $K$ be a finitely generated extension of
${\mathbb{F}}_{p}$. 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 $d,s\in{\mathbb{N}}$ 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 for a
given $K$ 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
$B\subseteq K^{m}$,

$|E\cap(A\times B)|\leq
C(|A|^{1-\epsilon}|B|^{\frac{d-1}{d}(1+\epsilon)}+|A|+|B|).$

###  1.1. 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, [19, Theorem 8.3]:

######  Fact 1.2.

For any finite $P$ and $L$,

$|\\{(p,l)\in P\times L:p\in l\\}|\leq
4|P|^{\frac{2}{3}}|L|^{\frac{2}{3}}+4|P|+|L|.$

Statements of the form of Theorem 1.1 can be seen as generalisations of this,
replacing the point-line incidence relation with other algebraic binary
algebraic relations. For characteristic 0 fields, such results were proven
first in [11, Theorem 9], and subsequently strengthened in [12, Theorem 1.2].
Using such bounds for binary relations, Elekes-Szabó [11] obtained analogous
bounds for ternary algebraic relations.

In positive characteristic, versions of Fact 1.2 have been proven ([3],[17])
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 ([3], [19, Theorem 2.55]).

Meanwhile, a special case of results in [4] and [7, Section 2] yields a
version of Theorem 1.1 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. Distality is a notion originating in model theory which
imposes a certain combinatorial restriction on the complexity of definable
sets in a structure; this notion and its incidence theoretic implications are
summarised in Section 3.1 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 1.1 by finding sufficient distality to trigger the incidence
bounds of [4]. 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. It follows from [15] 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 [14, Section 5], which suggests
a unifying explanation for all the results on existence of bounds described
above: they are all incarnations of 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, unboundedly large 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 algebraic part
${\mathbb{K}}\cap{\mathbb{F}}_{p}^{\operatorname{alg}}$, incidence bounds and
Elekes-Szabó results should go through as in characteristic 0.

We partially confirm this only in the special111See Proposition 4.2 case of
fields admitting finite residue field. However, in Theorem 7.1 we do confirm
for such $K$ that an Elekes-Szabó result applies: a mild strengthening of
Theorem 1.1 suffices as input to the proof of one of the main results of [2],
yielding Elekes-Szabó bounds for algebraic relations of arbitrary arity and
codimension in $K^{n}$ which do not arise from 1-dimensional algebraic groups.

###  1.2. 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. Thanks also to Martin
Hils for pointing out an error in the bounds in Remark 6.2 in an earlier
version of the paper.

_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._

##  2\. Preliminaries

We use basic notions and notation from model theory, as presented in e.g. [20,
Chapter 1].

Let $\mathcal{L}$ be a (possibly many-sorted) first order language and $T$ a a
complete $\mathcal{L}$-theory.

######  Notation 2.1.

If ${\mathcal{M}}\vDash T$ and $B\subseteq{\mathcal{M}}$ and
$x=(x_{1},\ldots,x_{n})$ is a tuple of variables of sorts
$S_{1},\ldots,S_{n}$, we write $B^{x}$ for $\prod_{i}(S_{i}({\mathcal{M}})\cap
B)$. We write $|x|$ for the length $|x|=n$ of the tuple.

For a set $B$, we write $B_{0}\subseteq_{{\operatorname{fin}}}B$ 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 $b\in{\mathcal{M}}^{x}$ and
$A\subseteq{\mathcal{M}}$, we set
${\operatorname{tp}}_{\phi}(b/A):=\\{\phi(x,c)^{\epsilon}:c\in
A^{y};\;\epsilon\in\\{0,1\\};\;{\mathcal{M}}\vDash\phi(b,c)^{\epsilon}\\}$.
The partitioning will often be left implicit.

##  3\. Distality

###  3.1. Distal cell decompositions

We recall the following definition from [4]:

######  Definition 3.1.

Let $A$ and $B$ be sets. A binary relation $E\subseteq A\times B$ admits a
distal cell decomposition with exponent $t\in{\mathbb{R}}$ if there exist
$s\in{\mathbb{N}}$ and finitely many relations $\Delta_{i}\subseteq A\times
B^{s}$ and $C\in{\mathbb{R}}$ such that for every
$B_{0}\subseteq_{{\operatorname{fin}}}B$, $A$ 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 is cut by no $E(b)$ for
$b\in B_{0}$, i.e. $\Delta_{i}(c)\subseteq E(b)$ or $\Delta_{i}(c)\cap
E(b)=\emptyset$.

It was proven in [4] that relations admitting distal cell decompositions enjoy
certain incidence bounds. For our purposes, the following version of this
deduced in [8, Theorem 2.6,2.7(2)] is most relevant.

A binary relation $E\subseteq A\times B$ is $K_{d,s}$-free if it contains no
subset $A_{0}\times B_{0}$ with $|A_{0}|=d$ and $|B_{0}|=s$.

######  Fact 3.2.

Let $E\subseteq A\times B$ be $K_{d,s}$-free and admit a distal cell
decomposition with exponent $t$. Then for
$A_{0}\subseteq_{{\operatorname{fin}}}A$ and
$B_{0}\subseteq_{{\operatorname{fin}}}B$,

$|E\cap(A_{0}\times B_{0})|\leq
O_{E}(|A_{0}|^{\frac{(t-1)d}{td-1}}|B_{0}|^{\frac{td-t}{td-1}}+|A_{0}|+|B_{0}|).$

###  3.2. Distal subsets

######  Definition 3.3.

Let ${\mathcal{M}}\vDash T$. Let $\phi(x;y)$ be an
$\mathcal{L}({\mathcal{M}})$ formula, and let $A,B\subseteq{\mathcal{M}}$ be
subsets.

  * •

An $\mathcal{L}$-formula $\zeta_{\phi}(x;z)$ is a uniform strong honest
definition (USHD) for $\phi$ on $A$ over $B$ if for any $a\in A$ and finite
subset $B_{0}\subseteq_{{\operatorname{fin}}}B$ with $|B_{0}|\geq 2$, there is
$d\in B_{0}^{z}$ such that
${\operatorname{tp}}(a/B_{0})\ni\zeta_{\phi}(x,d)\vdash{\operatorname{tp}}_{\phi}(a/B_{0})$.

  * •

We omit “on $A$” in the case $A={\mathcal{M}}$.

  * •

We omit “over $B$” in the case $B=A$.

  * •

$A$ is distal in ${\mathcal{M}}$ if every $\mathcal{L}(A)$-formula $\phi(x;y)$
has a USHD on $A$.

The notion of a strong honest definition comes from [6]. We work with USHDs
rather than directly with distal cell decompositions in order to be able to
reduce to one variable (Lemma 3.5), 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.

######  Remark 3.4.

An $\mathcal{L}$-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
$\mathcal{L}$-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

$\zeta(x,z_{1},\ldots,z_{s},w_{1},\ldots,w_{s},w^{\prime}_{1},\ldots,w^{\prime}_{s}):=\bigwedge_{i}(\delta_{i}(x,z_{i})\leftrightarrow
w_{i}=w^{\prime}_{i})$

is a USHD for $\phi$ on $A$ over $B$.

In particular, if $A\subseteq{\mathcal{M}}$ is distal in ${\mathcal{M}}$, then
the trace on $A$ of any $\mathcal{L}(A)$-formula $\phi(x,y)$ admits a distal
cell decomposition.

###  3.3. Reductions

######  Lemma 3.5.

A subset $A\subseteq{\mathcal{M}}$ is distal in ${\mathcal{M}}$ if and only if
any $\mathcal{L}$-formula $\phi(x;y)$ with $|x|=1$ has a USHD on $A$.

###### Proof.

First, it follows by an inductive argument from the 1-variable case that any
$\mathcal{L}$-formula has a USHD on $A$; we refer to the proof of [1,
Proposition 1.9] for this argument.

It remains to deduce that any $\mathcal{L}(A)$-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 $A$. ∎

######  Lemma 3.6.

Let ${\mathcal{M}}$ be an $\mathcal{L}$-structure. Let $S$ and $\widetilde{S}$
be $\mathcal{L}$-sorts and let $f:\widetilde{S}\rightarrow S$ be an
$\mathcal{L}$-definable function with uniformly finite fibres, say
$|f^{-1}(b)|\leq N$ for all $b\in f(S)$. Suppose $B\subseteq
f(S({\mathcal{M}}))$, and let
$\widetilde{B}:=f^{-1}(B)\subseteq\widetilde{S}$.

Let $A\subseteq{\mathcal{M}}^{x}$ and let $\phi(x,y)$ be an
$\mathcal{L}$-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 $B$.

###### Proof.

Say $\zeta(x,w)$ is a USHD for $\phi(x;f(z))$ over $\widetilde{B}$.

Let $B_{0}\subseteq_{{\operatorname{fin}}}B$ 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
${\operatorname{tp}}(a/\widetilde{B}_{0})\ni\zeta(x,\widetilde{d})\vdash{\operatorname{tp}}_{\phi(x;f(z))}(a/\widetilde{B}_{0})\vdash{\operatorname{tp}}_{\phi(x;y)}(a/B_{0})$.

Let $d:=f(\widetilde{d})$. Then $|f^{-1}(d)|\leq N^{|w|}$, and so there is
$M<N^{|w|}$ and $\overline{\epsilon}_{0}\in\\{0,1\\}^{M}$ and
${\overline{b}}_{0}\in(B_{0})^{M}$ such that
$\theta_{M,\overline{\epsilon}_{0}}(w,d,{\overline{b}}_{0})$ has the minimal
number of realisations amongst the formulas

$\theta_{n,\overline{\epsilon}}(w,d,{\overline{b}}):=(f(w)=d\wedge\forall
x.(\zeta(x,w)\rightarrow\bigwedge_{i=1}^{n}\phi(x,b_{i})^{\epsilon_{i}}))$

which hold of $\widetilde{d}$, with $n\in{\mathbb{N}}$ and
$\overline{\epsilon}\in\\{0,1\\}^{n}$ and ${\overline{b}}\in(B_{0})^{n}$. 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
$\theta_{M,\overline{\epsilon}_{0}}(w,d,{\overline{b}}_{0})\vdash\forall
x.(\zeta(x,w)\rightarrow\phi(x,b)^{\epsilon})$ for some
$\epsilon\in\\{0,1\\}$.

So ${\operatorname{tp}}(a/B_{0})\ni\exists
w.(\theta_{M,\overline{\epsilon}_{0}}(w,d,{\overline{b}}_{0})\wedge\zeta(x,w))\vdash{\operatorname{tp}}_{\phi(x;y)}(a/B_{0})$.
Coding the finitely many such formulas with $M<N^{|w|}$ and
$\overline{\epsilon}_{0}\in\\{0,1\\}^{M}$ into a single formula, we therefore
obtain a USHD for $\phi(x;y)$ on $A$ over $B$.

∎

######  Remark 3.7.

The finiteness assumption in Lemma 3.6 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 $x<_{o}x^{\prime}$ 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 $X_{0}\subseteq_{{\operatorname{fin}}}X$ and $o\in
O_{X}$, then ${\operatorname{tp}}_{=}(x/X_{0})$ is implied by the $<_{o}$-cut
of $x$ in $X_{0}$).

###  3.4. Remarks

We add some further remarks concerning these definitions, which will not be
used subsequently.

######  Remark 3.8.

Suppose $B$ is distal in an $\mathcal{L}$-structure ${\mathcal{M}}$. Then this
is expressed in the $\mathcal{L}_{P}$-theory of $({\mathcal{M}};B)$, where $P$
is a new predicate interpreted as $B$; i.e. if
$({\mathcal{M}}^{\prime};B^{\prime})\equiv({\mathcal{M}};B)$, then
$B^{\prime}$ is distal in ${\mathcal{M}}^{\prime}$.

######  Remark 3.9.

By [6, Theorem 21], ${\operatorname{Th}}({\mathcal{M}})$ is distal if and only
if ${\mathcal{M}}$ is distal in ${\mathcal{M}}$. (No saturation assumption is
needed here, thanks to Remark 3.8.)

######  Remark 3.10.

Distality in ${\mathcal{M}}$ of a subset $B\subseteq{\mathcal{M}}$ is
equivalent to distality of the induced structure $(B;(\phi(B)_{\phi\text{ an
${\mathcal{L}}$-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”.

######  Example 3.11.

If $B=(b_{i})_{i}\subseteq{\mathcal{M}}$ is an $\emptyset$-indiscernible
sequence which is not totally indiscernible, and this is witnessed by an
$\mathcal{L}$-formula $\theta_{<}$ with
${\mathcal{M}}\vDash\theta_{<}(b_{i},b_{j})\Leftrightarrow i<j$, then $B$ is
distal in ${\mathcal{M}}$.

######  Remark 3.12.

The argument of [6] to obtain uniformity of honest definitions goes through in
this setting. Namely, if $B$ is a subset of a model ${\mathcal{M}}$ of a
complete NIP $\mathcal{L}$-theory $T$, and the $\mathcal{L}_{P}$-structure
$({\mathcal{M}};B)$ is $|T|^{+}$-saturated, then $B$ is distal in
${\mathcal{M}}$ if and only if for any singleton $b\in B$ and any subset
$A\subseteq B$, ${\operatorname{tp}}^{{\mathcal{M}}}(b/A)$ is compressible in
the sense of [18]. This follows from a “$(p,q)$-argument” and transitivity of
compressibility.

It follows in particular that Example 3.11 can be generalised slightly when
${\mathcal{M}}$ is NIP: any $\emptyset$-indiscernible sequence which is not
totally indiscernible is distal in ${\mathcal{M}}$.

######  Question 3.13.

The following question was asked by Hrushovski and Pillay. By a result of
Simon, an NIP theory is distal if and only if every generically stable Keisler
measure is smooth. Does a version of this result go through for distality of
subsets of NIP structures? Is $B$ distal in ${\mathcal{M}}$ if and only if
every generically stable Keisler measure on
${\operatorname{Th}}_{\mathcal{L}_{P}}({\mathcal{M}},B)$ with $\mu(\neg P)=0$
is smooth? This might provide an alternative route to Theorem 5.6.

##  4\. Fields admitting valuations with finite residue field

By classical results in valuation theory, a valuation on a field $K$ can be
extended to any finite extension of $K$ with a finite extension of the residue
field [10, Theorem 3.1.2, Corollary 3.2.3], and can be extended to the
transcendental extension $K(X)$ without extending the residue field [10,
Corollary 2.2.3]. Since ${\mathbb{F}}_{p}$ and ${\mathbb{Q}}$ admit valuations
with finite residue field (respectively trivial and $p$-adic), we inductively
obtain:

######  Lemma 4.1.

Let $K$ be a finitely generated field. Then $K$ admits a valuation with finite
residue field.

If $K$ is a valued field of characteristic $p>0$, then the induced valuation
on the algebraic part $K\cap{\mathbb{F}}_{p}^{\operatorname{alg}}$ is trivial.
So a positive characteristic field which admits a valuation with finite
residue field has finite algebraic part. However, the converse fails.

######  Proposition 4.2.

For any prime $p$, there exists an algebraic extension
$L\geq{\mathbb{F}}_{p}(t)$ such that
$L\cap{\mathbb{F}}_{p}^{\operatorname{alg}}={\mathbb{F}}_{p}$ but no valuation
on $L$ has finite residue field.

###### Proof.

We work in an algebraic closure ${\mathbb{F}}_{p}(t)^{\operatorname{alg}}$ of
${\mathbb{F}}_{p}(t)$. Let
$\wp:{\mathbb{F}}_{p}(t)^{{\operatorname{alg}}}\rightarrow{\mathbb{F}}_{p}(t)^{\operatorname{alg}}$
be the Artin-Schreier map $\wp(x):=x^{p}-x$, an additive homomorphism with
kernel ${\mathbb{F}}_{p}$.

######  Claim.

$\deg({\mathbb{F}}_{p}(t,(\wp^{-1}(t^{a}))_{a>0})/{\mathbb{F}}_{p}(t))$ is
infinite.

###### Proof.

By [16, Theorem 8.3], it suffices to see that $\\{t^{a}|a>0\\}$ is not
contained in any finite union of additive cosets of
$\wp({\mathbb{F}}_{p}(t))$. Let $a,b\in{\mathbb{N}}\setminus p{\mathbb{N}}$ be
distinct. Let $\beta_{a,b}:=\sum_{i\geq
0}(t^{ap^{i}}-t^{bp^{i}})\in{\mathbb{F}}_{p}[[t]]$. Then
$\wp(\beta_{a,b})=t^{b}-t^{a}$. Now $\beta_{a,b}\notin{\mathbb{F}}_{p}(t)$,
since there are arbitrarily long intervals between exponents with non-zero
coefficient in this power series. So $(t^{a})_{a\in{\mathbb{N}}\setminus
p{\mathbb{N}}}$ lie in distinct cosets of $\wp({\mathbb{F}}_{p}(t))$. ∎

We write ${\operatorname{res}}$ for the residue field map associated to a
chosen valuation $v$ on a field $K$, and ${\operatorname{res}}(K)$ for the
corresponding residue field.

######  Claim.

Let $K^{\prime}\geq K\geq{\mathbb{F}}_{p}(t)$ be finite field extensions, and
suppose
$K^{\prime}\cap{\mathbb{F}}_{p}^{\operatorname{alg}}={\mathbb{F}}_{p}$. Let
$v$ be a valuation on $K$ with ${\operatorname{res}}(K)$ finite.

Then there exists a finite field extension $K^{\prime\prime}\geq K^{\prime}$
such that
$K^{\prime\prime}\cap{\mathbb{F}}_{p}^{\operatorname{alg}}={\mathbb{F}}_{p}$
but for any extension of $v$ to $K^{\prime\prime}$,
${\operatorname{res}}(K^{\prime\prime})\gneq{\operatorname{res}}(K)$.

###### Proof.

The valuation $v$ is non-trivial, so say $v(s)>0$. So $v$ induces the $s$-adic
valuation on ${\mathbb{F}}_{p}(s)\leq K$. Now $s$ is transcendental, so $t$ is
algebraic over $s$, so $K$ is also a finite extension of
${\mathbb{F}}_{p}(s)$. So we may assume without loss that $v$ restricts to the
$t$-adic valuation on ${\mathbb{F}}_{p}(t)$.

Since ${\operatorname{res}}(K)$ is finite, it is not Artin-Schreier closed;
say $\alpha\in{\operatorname{res}}(K)\setminus\wp({\operatorname{res}}(K))$.
Say ${\operatorname{res}}(\bar{\alpha})=\alpha$.

Since $\deg(K^{\prime}/{\mathbb{F}}_{p}(t))$ is finite, it follows from the
above Claim that

$\deg(K^{\prime}(\wp^{-1}(\bar{\alpha}),(\wp^{-1}(\bar{\alpha}+t^{a}))_{a>0})/K^{\prime})$

is infinite. So say $a>0$ is such that
$K^{\prime\prime}:=K(\wp^{-1}(\bar{\alpha}+t^{a}))\not\subseteq
K^{\prime}({\mathbb{F}}_{p^{p}})$. Then by considering degrees,
$K^{\prime\prime}\cap{\mathbb{F}}_{p}^{\operatorname{alg}}={\mathbb{F}}_{p}$.
But for any extension of $v$ to $K^{\prime\prime}$,

$\wp({\operatorname{res}}(\wp^{-1}(\bar{\alpha}+t^{a})))={\operatorname{res}}(\bar{\alpha}+t^{a})=\alpha\notin\wp({\operatorname{res}}(K)),$

so ${\operatorname{res}}(K^{\prime\prime})\gneq{\operatorname{res}}(K)$. ∎

Now we recursively construct a chain $K_{0}:={\mathbb{F}}_{p}(t)\leq
K_{1}\leq\ldots$ of finite extensions of ${\mathbb{F}}_{p}(t)$. Let
$\eta:\omega\times\omega\rightarrow\omega$ be a bijection such that
$\eta(i,j)\geq i$ for all $i,j$.

Note that ${\mathbb{F}}_{p}(t)$ 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
${\mathbb{F}}_{p}(s)\leq{\mathbb{F}}_{p}(t)$; there are only countably many
choices for $s$, and only finitely many ways to extend a valuation to a finite
extension ([10, Theorem 3.2.9]). 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 $K_{k}\geq
K_{i}\geq{\mathbb{F}}_{p}(t)$ and the valuation $v_{i,j}$ on $K_{i}$.

Now let $K_{\omega}:=\bigcup_{k<\omega}K_{k}$. We have
$K_{\omega}\cap{\mathbb{F}}_{p}^{\operatorname{alg}}={\mathbb{F}}_{p}$ since
this holds for each $K_{k}$.

Suppose $v$ is a valuation on $K_{\omega}$ with finite residue field. Then
${\operatorname{res}}(K_{\omega})={\operatorname{res}}(K_{i})$ say, and the
restriction of $v$ to $K_{i}$ is $v_{i,j}$ say. Then
${\operatorname{res}}(K_{\eta(i,j)+1})={\operatorname{res}}(K_{i})$,
contradicting the construction. ∎

######  Remark 4.3.

One might expect that a Zorn argument could replace the recursive construction
of the previous Proposition, i.e. that any maximal regular extension of
${\mathbb{F}}_{p}(t)$ has no valuation with finite residue field. But
${\mathbb{F}}_{p}((t^{\mathbb{Q}}))\cap{\mathbb{F}}_{p}(t)^{\operatorname{alg}}$
is a counterexample. Thanks to Zoé Chatzidakis for pointing this out.

##  5\. Distality in ${\operatorname{ACVF}}$ of subfields with finite residue
field

###  5.1. Uniform Swiss cheese decompositions

Let $L$ be a non-trivially valued algebraically closed field. Write $v$ for
the valuation map and ${\operatorname{res}}$ for the residue field map. We
consider $L$ as an
$\mathcal{L}_{\operatorname{div}}:=\\{+,-,\cdot,|,0,1\\}$-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 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 $\alpha\in
v(L)\cup\\{-\infty,+\infty\\}$.

######  Fact 5.1 (Canonical Swiss cheese decomposition [13, Theorem 3.26]).

Any boolean combination of balls can be represented as a finite disjoint union
of “Swiss cheeses”
$\dot{\bigcup}_{i<k}(b_{i}\setminus\dot{\bigcup}_{j<k_{i}}b_{ij})$, where the
$b_{i}$ are balls, each $b_{ij}$ is a proper sub-ball of $b_{i}$, for each $i$
the $b_{ij}$ are disjoint, and no $b_{i}$ is equal to any $b_{i^{\prime}j}$.
This representation is unique up to permutations.

We call the $b_{i}$ the “rounds” and the $b_{ij}$ the “holes” of a Swiss
cheese decomposition, and we say such a decomposition has complexity $\leq N$
if there are $k\leq N$ rounds each with $k_{i}\leq N$ holes. Let $\phi(x,y)$
be an $\mathcal{L}_{\operatorname{div}}$-formula with $|x|=1$. For any $a\in
L$, it follows directly from quantifier elimination that $\phi(L,a)$ is a
boolean combination of balls. We will need the following form of uniformity in
$a$ of the Swiss cheese decompositions.

######  Lemma 5.2.

There are $N$ and $d$ depending only on $\phi$ such that for all $a\in L^{y}$,
$\phi(L,a)$ has a Swiss cheese decomposition of complexity $\leq N$, each
round and each hole of which contains a point in a field extension of the
subfield generated by $a$ of degree dividing $d$.

###### Proof.

By quantifier elimination, $\phi(x,y)$ is equivalent to a boolean combination
of formulas of the form $\phi_{i}(x,y):=v(f_{i}(x,y))<v(g_{i}(x,y))$ for
polynomials $f_{i},g_{i}\in{\mathbb{Z}}[x,y]$.

Given $i$ and $a\in L$, let $\alpha_{j}$ resp. $\beta_{j}$ be the roots of
$f_{i}(x,a)$ resp. $g_{i}(x,a)$ in $L$. Then
$\phi_{i}(x,a)\Leftrightarrow\sum_{j}v(x-\alpha_{j})<\sum_{j}v(x-\beta_{j})$.

######  Claim 5.3.

$\phi_{i}(x,a)$ is a boolean combination of balls centred at the $\alpha_{j}$
and $\beta_{j}$.

###### Proof.

We show more generally, by induction on $s$, that any affine linear constraint
$\sum_{i=1}^{s}n_{i}v(x-\gamma_{i})<\nu$ is equivalent to a boolean
combination of balls centred at the $\gamma_{i}$. Conditioning on the finitely
many cases, it suffices to show this for a given order type of
$\epsilon:=v(x-\gamma_{1})$ over $\\{v(\gamma_{1}-\gamma_{i}):i>1\\}$. If
$\epsilon=v(\gamma_{1}-\gamma_{i})$ for some $i>1$, we conclude by the
inductive hypothesis. Otherwise, by the ultrametric triangle inequality,
$v(x-\gamma_{i})=\epsilon$ if $\epsilon<v(\gamma_{1}-\gamma_{i})$, and
$v(x-\gamma_{i})=v(\gamma_{1}-\gamma_{i})$ otherwise, so the affine constraint
is equivalent to $\epsilon<\nu^{\prime}$ or $\epsilon>\nu^{\prime}$ for some
$\nu^{\prime}$. ∎

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

$d:={\operatorname{lcm}}_{i}({\operatorname{lcm}}(\deg_{x}f_{i},\deg_{x}g_{i})),$

and it follows that the rounds and holes in the Swiss cheese decomposition
also have this property and are bounded in number. ∎

###  5.2. 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 $\mathcal{L}^{{\operatorname{eq}}}$)
element relation $(\in)\subseteq L\times B$.

Given $N\in{\mathbb{N}}$, 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

$X_{N}:=\\{(b_{i})_{i<N},(b_{ij})_{i,j<N}:b_{i},b_{ij}\in B\textrm{ are as in
Fact~{}\ref{f:swiss}}\\}\subseteq B^{N(N+1)}$

we obtain a $\emptyset$-definable surjection
$f_{S_{N}}:X_{N}\twoheadrightarrow S_{N}$ defined by
$f_{S_{N}}((b_{i})_{i},(b_{ij})_{ij}):=[{\textrm{code of
}\bigcup_{i}(b_{i}\setminus\bigcup_{j}b_{ij})}]$. By Fact 5.1, any $c\in
S_{N}$ has a unique-up-to-permutation representation as a Swiss cheese
decomposition of complexity $\leq N$, so $f_{S_{N}}$ has finite fibres.

With a view to proving Theorem 5.6, for $K$ a valued subfield of $L$ with
finite residue field and $d\in{\mathbb{N}}$, define $B_{K,d}\subseteq B$ to be
the set of balls which contain an element of some finite field extension of
$K$ within $L$ of degree dividing $d$ over $K$. Let $X_{N}(B_{K,d}):=X_{N}\cap
B_{K,d}^{N(N+1)}$ and $S_{N}(B_{K,d}):=f_{S_{N}}(X_{N}(B_{K,d}))$.

######  Lemma 5.4.

Let $N,d\in{\mathbb{N}}$.

  1. (i)

$x\in y$ has a USHD over $B_{K,d}$.

  2. (ii)

$x\in f_{S_{N}}(y)$ has a USHD over $B_{K,d}$.

  3. (iii)

$x\in z$ has a USHD over $S_{N}(B_{K,d})$.

###### Proof.

  1. (i)

By assumption, the residue field of $K$ is a finite field, say
${\mathbb{F}}_{q}$.

Let $B_{0}\subseteq_{{\operatorname{fin}}}B_{K,d}$. Let
$B_{0}^{\prime}:=\\{b\vee b^{\prime}:b,b^{\prime}\in B_{0}\\}$ where the join
$b\vee b^{\prime}$ is the smallest ball containing both $b$ and $b^{\prime}$.
By the ultrametric triangle inequality, $B_{0}^{\prime}$ is then closed under
join. Note that $B_{0}^{\prime}\subseteq B_{K,d}$, since $B_{K,d}$ is upwards-
closed.

Let $p\in L$. Let $b\in B_{0}^{\prime}\cup\\{L\\}$ be minimal such that $p\in
b$, and let $b_{1},\ldots,b_{s}\in B_{0}^{\prime}$ be the maximal proper
subballs (if any) of $b$ in $B_{0}^{\prime}$. Then

$(x\in b\wedge\bigwedge_{i=1}^{s}x\notin b_{i})\vdash{\operatorname{tp}}_{x\in
y}(p/B_{0}),$

and each $b_{i}$ is the join of two balls in $B_{0}$, and either the same goes
for $b$ or $b=L$. So coding the finitely many possibilities yields a USHD as
required if we can bound $s$ independently of $p$.

Assume $s>1$. 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
$i\mapsto\lambda_{i}:={\operatorname{res}}(\frac{p_{i}-p_{1}}{p_{2}-p_{1}})$
is an injection of $\\{1,\ldots,s\\}$ into ${\operatorname{res}}(L)$. Indeed,
if $\lambda_{i}=\lambda_{j}$ then
${\operatorname{res}}(\frac{p_{i}-p_{j}}{p_{2}-p_{1}})=0$, 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 $d^{3}$, by the valuation inequality ([10, Corollary 3.2.3]).
each $\lambda_{i}$ is in an extension of
${\operatorname{res}}(K)={\mathbb{F}}_{q}$ of degree $\leq d^{3}$, so
$\lambda_{i}\in{\mathbb{F}}_{q^{d^{3}!}}$ for all $i$. So $s\leq q^{d^{3}!}$.

  2. (ii)

$x\in f_{S_{N}}(y)$ 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 $B_{K,d}$.

  3. (iii)

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 3.6.

∎

###  5.3. Concluding distality

######  Lemma 5.5.

Let $L$ be a non-trivially valued algebraically closed field. Let $K\leq L$ be
a subfield and suppose ${\operatorname{res}}(K)$ is finite. Let $\phi(x;y)$ be
an $\mathcal{L}_{\operatorname{div}}$-formula with $|x|=1$. Then $\phi$ has a
USHD over $K$.

Moreover, for any $r\geq 1$, $\phi$ has a USHD over the set $K_{r}\subseteq
K^{{\operatorname{alg}}}\subseteq L$ of points with degree over $K$ dividing
$r$,

$K_{r}:=\\{a\in L:\deg(K(a)/K)|r\\}$

###### Proof.

Let $N$ and $d$ be as in Lemma 5.2 for $\phi$. Then there is a
$\emptyset$-definable function $h:L^{y}\rightarrow S_{N}$ such that
$L^{{\operatorname{eq}}}\vDash\forall x,y.(\phi(x,y)\leftrightarrow x\in
h(y))$, and $h(K_{r})\subseteq S_{N}(B_{K,dr})$.

By Lemma 5.4(iii), say $\zeta(x,z^{\prime}_{1},\ldots,z^{\prime}_{s})$ is a
USHD for $x\in y$ over $S_{N}(B_{K,dr})$. Then (an
$\mathcal{L}_{\operatorname{div}}$-formula equivalent to)
$\zeta(x,h(z_{1}),\ldots,h(z_{s}))$ is a USHD for $\phi(x;y)$ over $K_{r}$. ∎

######  Theorem 5.6.

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 5.5.

###### 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 5.5 and Lemma 3.5. ∎

######  Remark 5.7.

This does not reprove distality of ${\mathbb{Q}}_{p}$, because
${\mathbb{Q}}_{p}$ does not eliminate quantifiers in
$\mathcal{L}_{\operatorname{div}}$.

##  6\. Incidence theory consequences

######  Theorem 6.1.

Let $K$ be a valued field with finite residue field. Let $E\subseteq
K^{n}\times K^{m}$ be quantifier-free definable in
$\mathcal{L}_{\operatorname{div}}(K)$. Suppose $E$ omits $K_{d,s}$, where
$d,s\in{\mathbb{N}}$. Then there exist $t$ (see Remark 6.2) and $C>0$ such
that for $A_{0}\subseteq_{{\operatorname{fin}}}K^{n}$ and
$B_{0}\subseteq_{{\operatorname{fin}}}K^{m}$,

$|E\cap(A_{0}\times B_{0})|\leq
C(|A_{0}|^{\frac{(t-1)d}{td-1}}|B_{0}|^{\frac{td-t}{td-1}}+|A_{0}|+|B_{0}|).$

The same holds if $K$ is replaced by $K_{r}\subseteq K^{{\operatorname{alg}}}$
defined as in Lemma 5.5.

###### Proof.

By Theorem 5.6 and Remark 3.4, $E$ admits a distal cell decomposition, and we
conclude by Fact 3.2. ∎

The version of this stated in the introduction, Theorem 1.1, follows by
considering Lemma 4.1 and the special case that $E$ is defined as the zero set
of polynomials over $K$, and setting $\epsilon:=\frac{1}{dt-1}$.

######  Remark 6.2.

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 $q=|{\operatorname{res}}(K)|^{r}$ and $d$ is as in the
proof of Lemma 5.5. Indeed, this is the exponent arising from bounding the
number of balls used in Lemma 5.4(i), and neither Lemma 5.4(ii) nor Lemma 3.6
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 3.2. 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 [21],
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.

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 $t:=4(q+1)$.

The proof of Lemma 5.4(i) in this case gives a USHD
$\zeta(y,x,{\overline{z}})$ 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 [21, Theorem 2.1], if an $\mathcal{L}_{\operatorname{div}}$ qf-formula
$\psi(x,{\overline{y}},{\overline{z}})$ is linear in $x,{\overline{y}}$, i.e.
each polynomial has degree 1 in $x$ and each $y_{i}$, then $\exists
x.\psi(x,{\overline{y}},{\overline{z}})$ is equivalent modulo
${\operatorname{ACVF}}$ to a qf-formula linear in ${\overline{y}}$.

Now the formula $\zeta(y,x,{\overline{z}})\rightarrow(x,y)Ew$ is linear in
$x,y$, so $\forall y.(\zeta(y,x,{\overline{z}})\rightarrow(x,y)Ew)$ is
equivalent to a qf-formula which is linear in $x$. Similarly $\forall
y.(\zeta(y,x,{\overline{z}})\rightarrow\neg(x,y)Ew)$ 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 $\xi(x,{\overline{w}})$ over $K$ of exponent $\leq
2(q+1)$. Then $\xi(x,{\overline{w}})\wedge\zeta(y,x,{\overline{z}})$ is a USHD
for $E$ over $K$ of exponent $\leq 2(q+1)+2(q+1)=4(q+1)$. In symmetric form,
this gives a bound of $O(N^{\frac{3}{2}-\frac{1}{16(q+1)-2}})$ on the number
of incidences of $N$ lines and $N$ points.

As a final remark, note that although Theorem 6.1 does apply in characteristic
0, e.g. to $K={\mathbb{Q}}_{p}$, the bounds we obtain in this way are worse
than those obtained from [12] by embedding ${\mathbb{Q}}_{p}$ in
${\mathbb{C}}={\mathbb{R}}^{2}$, even for $p=2$.

######  Question 6.3.

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 $X,A\subseteq{\mathbb{F}}_{p}(t)^{2}$ we have $|\\{((x,y),(a,b))\in
X\times A:y=ax+b\\}|\leq C\max(|X|,|A|)^{\frac{3}{2}-\epsilon}$? (Remark 6.2
yields a bound depending on $p$ of $\epsilon=\frac{1}{16p+14}$ in this case.
Meanwhile one can obtain a lower bound exponent of $\frac{4}{3}$ by
considering a rectangular example with bounded degree polynomials,
${\mathbb{F}}_{p}[t]_{<n}\times{\mathbb{F}}_{p}[t]_{<2n}$.)

##  7\. Elekes-Szabó consequences

Elekes-Szabó [11] exploit incidence bounds in characteristic zero to find that
commutative algebraic groups are responsible for ternary algebraic relations
with asymptotically large intersections with finite grids. In [2], this is
generalised to relations of arbitrary arity. In this section, we remark that
these arguments go through in the present positive characteristic context, at
least if we restrict to the 1-dimensional situation of [2, Theorem 1.4].

Let $K_{0}$ be a field admitting a valuation with finite residue field (e.g. a
function field over a finite field). Let ${\mathcal{U}}$ be a non-principal
ultrafilter on $\omega$. Define

$K^{\prime}:=((K_{0})^{\mathcal{U}})^{\operatorname{alg}}\leq((K_{0})^{\operatorname{alg}})^{\mathcal{U}}=:K.$

For $r\geq 1$, let

$K_{r}:=\\{a\in(K_{0})^{\operatorname{alg}}:\deg(K_{0}(a)/K_{0})|r\\}.$

So (by Łoś’s theorem) we have
$K^{\prime}=\bigcup_{r\in\omega}(K_{r})^{\mathcal{U}}$.

We work with the setup of [2, 2.1], with $(K_{0})^{\operatorname{alg}}$ in
place of ${\mathbb{C}}$, and in a countable language in which each of these
internal sets $(K_{r})^{\mathcal{U}}\subseteq K$ is definable.

######  Theorem 7.1.

Let $K_{0}$ be a field admitting a valuation with finite residue field. Let
$V\subseteq{\mathbb{A}}^{n}$ be an affine algebraic variety defined over
$K_{0}$ of dimension $d$. Then at least one of the following holds:

  1. (i)

$V$ admits a powersaving on $K_{0}$: there exist $C,\epsilon>0$ such that for
all $X_{i}\subseteq_{{\operatorname{fin}}}K_{0}$, $i=1,\ldots,n$, we have

$|V(K_{0})\cap\prod_{i}X_{i}|\leq C(\max_{i}|X_{i}|)^{d-\epsilon}.$

  2. (ii)

$V$ is _special_ : $V$ is in co-ordinatewise correspondence222As defined in
[2, Definition 1.1] 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.

###### Proof.

Let $K^{\prime}\leq K$ be as above. Also let $C_{0}\leq K^{\prime}$ be a
countable algebraically closed subfield over which $V$ is defined.

The proof in [2] goes through, but using Theorem 6.1 in place of [2, Theorem
2.14], and with [9, Theorem 3.3.1] replacing [2, Proposition A.4]. We describe
the necessary changes.

Firstly, [2, Theorem 2.15] goes through in the case that
$X_{i}\subseteq((K_{r})^{\mathcal{U}})^{n_{i}}$ for some $r$ ($i=1,2$). The
proof is identical, using Theorem 6.1; the sublinearity of the dependence on
$s$ where $K_{2,s}$ is omitted, discussed after [2, Theorem 2.14], also holds
here: this is described in [8, Remark 2.7(2),Corollary 2.8], and is proven
explicitly in [5, Theorem 2.6]. (In fact this sublinearity isn’t necessary for
the present 1-dimensional case.)

Now [2, Theorem 5.9] goes through for $P\subseteq(K^{\prime})^{<\omega}$. The
proof is identical, except that in the proof of [2, Proposition 5.14], since
${\overline{a}},{\overline{d}}\in(K^{\prime})^{<\omega}$, already
${\overline{a}},{\overline{d}}\in((K_{r})^{\mathcal{U}})^{<\omega}$ for some
$r$, and this passes through to the types $X_{i}$ since
$(K_{r})^{\mathcal{U}}$ is definable, so the above restricted form of [2,
Theorem 2.15] applies.

Next, ${\operatorname{End}}^{0}_{C_{0}}(G)$ must be redefined as the skew-
field of quotients of ${\operatorname{End}}_{C_{0}}(G)$ (this agrees with
${\mathbb{Q}}\otimes{\operatorname{End}}_{C_{0}}(G)$ in characteristic 0); see
[9, 3.1] for discussion of the possibilities.

Finally, we indicate how to circumvent the use of [2, Proposition A.4], which
is proven only in characteristic 0, in the 1-dimensional case. Where this is
applied in [2, Proposition 6.1], we have $a_{i}\in K^{\prime}$
($i=1,\ldots,n$) such that
${\mathcal{G}}_{\overline{a}}=\\{{\operatorname{acl}}^{0}(a_{i}):i\\}$ embeds
in a projective subgeometry of the ${\operatorname{acl}}^{0}$-geometry
${\mathcal{G}}_{K}$ of $K$. (Here we have $a_{i}\in K^{\prime}$ rather than
$a_{i}\in K^{<\omega}$, as this is what arises in the proof, via [2, Theorem
7.4], in the 1-dimensional case corresponding to the statement of the current
theorem.) By [9, Theorem 3.3.1], there is a 1-dimensional algebraic group $G$
over $C_{0}$ and generic ${\overline{x}}\in G^{m}$ over $C_{0}$ (where
$m=\dim({\mathcal{G}}_{\overline{a}})$) and
$A\in\operatorname{Mat}_{n,m}({\operatorname{End}}(G))$ such that, setting
${\overline{h}}:=A{\overline{x}}$, we have
${\operatorname{acl}}^{0}(h_{i})={\operatorname{acl}}^{0}(a_{i})$. Then
$\operatorname{loc}^{0}({\overline{h}})=AG^{m}$ is a connected algebraic
subgroup of $G^{n}$, as required.

The rest of the proof goes through unchanged. ∎

######  Remark 7.2.

The only obstruction to pushing this to higher dimension, i.e. to a version of
[2, Theorem 1.11], is the need to generalise the higher dimensional version of
Evans-Hrushovski [2, Proposition A.4] 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 [2, Proposition 7.10]; this may not be so easy to generalise,
and the statement may need to change.

For these reasons, we leave positive characteristic analogues of [2, Theorem
1.11] to future work.

## References

  * ACGZ [20] Matthias Aschenbrenner, Artem Chernikov, Allen Gehret, and Martin Ziegler, _Distality in valued fields and related structures_ , arXiv:2008.09889.
  * BB [18] Martin Bays and Emmanuel Breuillard, _Projective geometries arising from Elekes-Szabó problems_ , arXiv:1806.03422.
  * BKT [04] J. Bourgain, N. Katz, and T. Tao, _A sum-product estimate in finite fields, and applications_ , Geom. Funct. Anal. 14 (2004), no. 1, 27–57. MR 2053599
  * CGS [20] Artem Chernikov, David Galvin, and Sergei Starchenko, _Cutting lemma and Zarankiewicz’s problem in distal structures_ , Selecta Math. (N.S.) 26 (2020), no. 2, Paper No. 25, 27. MR 4079189
  * [5] Artem Chernikov, Ya’acov Peterzil, and Sergei Starchenko, _Model-theoretic Elekes-Szabó for stable and o-minimal hypergraphs_ , Forthcoming.
  * CS [15] Artem Chernikov and Pierre Simon, _Externally definable sets and dependent pairs II_ , Trans. Amer. Math. Soc. 367 (2015), no. 7, 5217–5235. MR 3335415
  * CS [18] Artem Chernikov and Sergei Starchenko, _A model-theoretic generalization of the Elekes-Szabó theorem_ , arXiv:1801.09301.
  * CS [20] by same author, _Model-theoretic Elekes-Szabó in the strongly minimal case_ , arXiv:1801.09301.
  * EH [91] David M. Evans and Ehud Hrushovski, _Projective planes in algebraically closed fields_ , Proc. London Math. Soc. (3) 62 (1991), no. 1, 1–24. MR 1078211
  * EP [05] Antonio J. Engler and Alexander Prestel, _Valued fields_ , Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005. MR MR2183496 (2007a:12005)
  * ES [12] György Elekes and Endre Szabó, _How to find groups? (and how to use them in Erdös geometry?)_ , Combinatorica 32 (2012), no. 5, 537–571. MR 3004808
  * FPS+ [17] Jacob Fox, János Pach, Adam Sheffer, Andrew Suk, and Joshua Zahl, _A semi-algebraic version of Zarankiewicz’s problem_ , J. Eur. Math. Soc. (JEMS) 19 (2017), no. 6, 1785–1810. MR 3646875
  * Hol [95] Jan E. Holly, _Canonical forms for definable subsets of algebraically closed and real closed valued fields_ , J. Symbolic Logic 60 (1995), no. 3, 843–860. MR 1348997
  * Hru [13] Ehud Hrushovski, _On pseudo-finite dimensions_ , Notre Dame J. Form. Log. 54 (2013), no. 3-4, 463–495. MR 3091666
  * KSW [11] Itay Kaplan, Thomas Scanlon, and Frank O Wagner, _Artin-Schreier extensions in NIP and simple fields_ , Israel Journal of Mathematics 185 (2011), no. 1, 141–153.
  * Lan [02] Serge Lang, _Algebra_ , third ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. MR MR1878556 (2003e:00003)
  * SdZ [17] Sophie Stevens and Frank de Zeeuw, _An improved point-line incidence bound over arbitrary fields_ , Bull. Lond. Math. Soc. 49 (2017), no. 5, 842–858. MR 3742451
  * Sim [20] Pierre Simon, _Type decomposition in NIP theories_ , J. Eur. Math. Soc. (JEMS) 22 (2020), no. 2, 455–476. MR 4049222
  * TV [06] Terence Tao and Van Vu, _Additive combinatorics_ , Cambridge Studies in Advanced Mathematics, vol. 105, Cambridge University Press, Cambridge, 2006. MR 2289012
  * TZ [12] Katrin Tent and Martin Ziegler, _A course in model theory_ , Lecture Notes in Logic, vol. 40, Association for Symbolic Logic, La Jolla, CA; Cambridge University Press, Cambridge, 2012. MR 2908005
  * Wei [84] Volker Weispfenning, _Quantifier elimination and decision procedures for valued fields_ , Models and sets (Aachen, 1983), Lecture Notes in Math., vol. 1103, Springer, Berlin, 1984, pp. 419–472. MR 775704