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\begin{document}
\title{Incidence bounds in positive characteristic via valuations and distality}
\author{Martin Bays}
\date{20.05.2021}
\maketitle
%Abstract:
% Joint work with \textbf{Jean-François Martin}.
% The Szemerédi-Trotter theorem bounds the number of incidences between
% finite sets of points and straight lines in the real plane, and
% generalisations to other algebraic binary relations play an important
% role in understanding the interaction between (pseudo)finite sets and
% field structure in zero characteristic. In positive characteristic,
% these incidence bounds fail drastically without further restrictions,
% but may be expected to hold under certain conditions. We confirm this in
% the case of fields admitting a valuation with finite residue field, e.g.
% finitely generated extensions of $\F_p$, by seeing that the restricted
% distality provided by the valuation suffices to trigger a result of
% Chernikov-Galvin-Starchenko.
% .
This talk is based on joint work with \textbf{Jean-François Martin}.
\tableofcontents
# Background
## Incidence bounds in characteristic zero
***)
Factn[Szemerédi-Trotter '83]:
There exists $C (- \R$ such that,
given $N$ points and $N$ lines in $\R^2$,
the number of incidences is bounded as
$$ |{ \{ (p,l) : p (- l \} }| <= C(N^{\frac32 - \frac16}) .$$
.
** This has been generalised in various ways. In particular:
Factn[Elekes-Szabó '12]:
If $(C_b)_{b (- B}$ is an algebraic family of distinct
irreducible plane curves over a field $K$ of characteristic 0,
there are $C,\eps > 0$ such that
given $N$ points in $K^2$ and $N$ curves in the family,
$$ |{ \{ (a,b) : a (- C_b \} }| <= C(N^{\frac32-\eps}).$$
.
** Hrushovski: this indicates a certain /modularity/ of the interaction
between (pseudo)finite sets and field structure.
** In particular, abelian groups are the only source of relations on which
finite sets ``maximally accumulate'':
Factnb[Elekes-Szabó '12 ($m=3$), Raz-Sharir-de Zeeuw '18 ($m=4$), B-Breuillard '20]:
Let $V (= \A^m$ be an irreducible affine variety over a field $K$ of characteristic 0.
Exactly one of the following holds:
**(i) $\exists C,\eps > 0.\;\forall X_1,...,X_m (=f K.$
$$| V(K) \cap (X_1\times...\times X_m) | <= C\max(|X_i|)^{\dim V - \eps};$$
** OR: up to finite correspondences on co-ordinates and taking products,
\\\ $V$ is a subgroup of a power of a 1-dimensional algebraic group.
*ee*
.
*e)*
## Incidence bounds in positive characteristic
***) In positive characteristic, these bounds utterly fail:
Remarkn:
Let $K := \F_p^{\alg}$.
For any algebraic set $V (= K^n$,
there is $r>0$ such that for arbitrarily large $n$,
$$|V(\|F_{p^n})| >= r(p^n)^{\dim V}.$$
(This follows from the Lang-Weil estimates.)
.
** However, Hrushovski conjectures that the Zilber trichotomy applies:
\\\ infinite pseudofinite fields should be the only obstruction to modularity.
\\\ (Above it is $\prod_{n --> \U} \|F_{p^n} <= K^\U$.)
\textit{(In the case of the sum-product theorem, this is true,
i.e. failures of sum-product bounds are due to finite subfields
(Bourgain-Katz-Tao, Tao-Vu, Tao, Hrushovski, Wagner).)}
** As an extreme case, this conjecture suggests that for $K$ with
\textbf{finite} algebraic part $K \cap \F_p^{\alg}$,
e.g. $K = \F_p(t)$,
the characteristic 0 results should go through.
*e)*
## Distality
***)
Definitionn:
Let $\M$ be an $\Lang$-structure.
***)
Let $\phi(x;y)$ be an $\Lang$ formula, let $A,B (= \M$.
An $\Lang$-formula $\zeta_\phi(x;z)$ is a \defn{uniform strong honest
definition (USHD)} for $\phi$ on $A$ over $B$ if
for any $a (- A^x$ and finite subset $B_0 (=f B$ with $|B_0| >= 2$,
there is $d (- B_0^z$
such that
$$\tp(a/B_0) -) \zeta_\phi(x,d) |- \tp_\phi(a/B_0)$$
(where
$\tp_\phi(a/B_0) := \{ \phi(x,b)^\eps (- \tp(a/B_0) : b (- B_0^y \}$).
** If $A = \M$, we omit ``on $A$''.
** $B (= \M$ is \defn{distal in $\M$} if every $\Lang$-formula $\phi(x;y)$
has a USHD on $B$ over $B$.
*e)*
.
** Reduction to one variable:
Factn{distal1}:
$B (= \M$ is distal in $\M$ iff any $\Lang$-formula $\phi(x;y)$ with $|x|
= 1$ has a USHD on $B$ over $B$.
.
Proofidea:
Given $a,b (- B$ and $\phi(x,y;z)$ and $B_0 (=f B$,
\\\say $\tp(a/bB_0) -) \zeta(x,b,c) |- \tp_{\phi(x;y,z)}(a/bB_0)$ with $c (- B_0^w$.
Now say $\tp(b/B_0) -) \theta(y,c') |- \tp_{\forall x. (\zeta(x,y,w) ->
\phi(x,y,z))}(b/B_0)$.
Then $\tp(ab/B_0) -) \zeta(x,y,c) /\ \theta(y,c') |- \tp^+_{\phi(x,y;z)}(ab/B_0)$.
Now repeat with $\neg\phi$.
Uniformity follows through.
.
**
Examplen:
If $B = (b_i)_i (= \M$ is a ${}$-indiscernible sequence,
\\\and there is an $\Lang$-formula $\theta_<$ with $\M |=
\theta_<(b_i,b_j) <=> i 0$ such that
% $$|\phi(X_0;Y_0)| <= O_\phi(N^{\frac32 - \eps}).$$
.
*e)*
# The result
## Statements
***)
Theorem{main}:
Let $k$ be a valued field with finite residue field.
\\\Then $k$ is distal in $k^{\alg} |= \"ACVF$.
.
(Note: a positive characteristic valued field with finite residue field is not distal as a structure, nor even NIP (by Kaplan-Scanlon-Wagner).)
**
So by Fact~{distInc}:
Corollaryn:
Let $k$ be a valued field with finite residue field.
Let $E (= k^n \times k^m$ be quantifier-free definable in $\Ldiv(k)$.
Suppose $E$ is $K_{d,s}$-free, where $d,s (- \N$.
Then there exist $t,C>0$ such that
for $A_0 (=f k^n$ and $B_0 (=f k^m$,
$$|E \cap (A_0 \times B_0)| <= C(|A_0|^{\frac{(t-1)d}{td-1}}
|B_0|^{\frac{td-t}{td-1}} + |A_0| + |B_0|).$$
.
** In particular this yields a Szemerédi-Trotter-style result:
Corollaryn:
Suppose $(C_b)_{b (- B (= k^m}$ is an algebraic family of distinct
irreducible plane curves over a field $k$ which admits a valuation with finite residue field.
Then $E := \{ (a,b) : a (- C_b(k) \} (= k^{2+m}$ is $K_{2,s}$-free for some
$s (- \N$.
So there exist $\eps_0,C>0$ such that for $A_0 (=f k^2$ and $B_0 (=f k^m$,
$$|E \cap (A_0 \times B_0)|
<= C(|A_0|^{1-\eps_0} |B_0|^{\frac12(1+\eps_0)} + |A_0| + |B_0|)$$
($\eps_0 := \frac1{2t-1} > 0$), or in symmetric form:
$$|E \cap (A_0 \times B_0)| <= C'(\max(|A_0|,|B_0|))^{\frac32-\eps}$$
($\eps := \frac{\eps_0}2 > 0$, $C' := 3C$).
.
%**
%Moreover, it will follow from the proof that $t$ in Fact~{distInc} can be
%bounded in terms of the size of the residue field.
*e)*
## Fields admitting finite residue field
Examplen:
The $t$-adic valuation on $\F_p(t)$ has residue field $\F_p$.
Similarly, any finitely generated extension of $\F_p$, i.e. any function
field over a finite field, admits a valuation with finite residue field.
%(e.g. $\F_q(s,t,\sqrt{s+t}) (= \F_q((s))((t))^{\alg}$.)
.
However:\
Proposition:
For any prime $p$,
there exists an algebraic extension $L >= \F_p(t)$ such that $L\cap
\F_p^\alg = \F_p$ but no valuation on $L$ has finite residue field.
.
Such an $L$ can be built by recursively adjoining Artin-Schreier roots which
force Artin-Schreier extensions of the residue fields of valuations on
previously built fields; using the Artin-Schreier version of Kummer theory,
one can always do this without extending the algebraic part.
# Proof of Theorem~{main}
***) Let $L |= \"ACVF$ and let $k (= L$ be a subfield with $\res(k)$ finite.
** We want to see that $k (= L$ is distal in $L$.
** By reduction to 1 variable, it suffices to see:
\\\any $\phi(x,y)$ with $|x| = 1$ has a USHD over $k$.
*e)*
## Compressing balls
***) By QE, for $a (- k$, $\phi(L,a)$ is a boolean combination of open and closed balls
$$v(x-a') > \alpha \text{ or } v(x-a') >= \alpha$$
centred at points $a'$ with $\deg(k(a')/k)$ bounded, say dividing $d$.
** Let $B_{k,d}$ be the set of balls (closed and open) centred at points of
degree $|d$ field extensions of $k$ within $L$.
Using the bounded size of the residue field of such extensions, we obtain:
Lemman:
$x (- y$ has a USHD over $B_{k,d}$.
.
Proof:
***) Let $a (- L$ and $B_0 (=f B_{k,d}$.
** Then $x (- (b \\ \bigcup_{i~~ 1$.
** Say $p_i (- b_i$ is of degree $|d$ over $k$, and let $\alpha (- v(L)$ be the radius of $b$.
** Now $$i |-> \lambda_i := \res({ \frac{p_i-p_1}{p_2-p_1} })$$
is an injection of $\{1,...,s\}$ into $\res(L)$:
***) If $i/=j$ then $b_i \/ b_j = b$, so $v(p_i-p_j)=\alpha$.
** Now suppose $\lambda_i=\lambda_j$.
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 $i=j$.
*e)*
** Say $\res(k) = \F_q$.
** Since each $\lambda_i$ is in the residue field of an extension of $k$ of
degree $|d^3$, by the valuation inequality
$$\lambda_i (- \F_{q^{d^3}}.$$
** So $s <= q^{d^3}$.
*e)*
.
** But this is not enough on its own.
To show that $\phi(x,y)$ has a USHD over $k$:
given $C (=f k$ we have to determine $\tp_\phi(a/C)$ using only $C$ as
parameters -- but the balls involved will generally not be defined over $C$!
** So we need to be more careful with the QE.
*e)*
## Compressing cheeses
***) $\phi(L,a)$ has a unique-up-to-permutation ``Swiss cheese decomposition''
as a finite union of disjoint cheeses,
$$\phi(L,a) = \bigcup_i (b_i \\ \bigcup_j b_{ij}),$$
where each cheese is a ball $b_i$ minus a finite union of disjoint proper subballs $b_{ij}$,
\\\and no $b_i$ is equal to any hole $b_{i'j}$.
** Moreover:
Factn[Uniform Swiss Cheese Decomposition]:
There are $N$ and $d$ depending only on $\phi$ such that for all $a (- L^y$,
$\phi(L,a)$ has Swiss cheese decomposition involving $<=N$ balls, where each
ball contains a point in a degree $|d$ field extension of the subfield
generated by $a$.
.
** Increasing $N$ and allowing the empty ball and its complement, we can
assume the form of the decomposition is constant, given by a Boolean term $D$.
** Let $X (= B^N$ be the set defined by the inclusion relations required for
$D(b_1,...,b_N)$ to be a Swiss cheese decomposition.
** So $D : X --> \text{[codes for subsets]}$ is definable with boundedly
finite fibres,
and
$$D(X \cap (B_{k,d})^N) )= \{ \ulcorner \phi(x,c) \urcorner : c (- k^y \}.$$
*e)*
## Collapsing USHDs
***) We conclude by a general elementary model theoretic lemma on USHDs:
Lemman:
Let $f$ be definable with boundedly finite fibres.
Let $C (= \"im(f)$.
If $\psi(x,f(z))$ has a USHD over $f^{-1}(C)$,
\\\then $\psi(x,y)$ has a USHD over $C$.
.
!newtup z
(Explicitly, to conclude:
***) apply this Lemma to $x (- D(\z)$, which has a USHD over $B_{k,d}$ by ball
compression;
** this yields that $x (- w$ has a USHD over
$$\{ \ulcorner \phi(x,c) \urcorner : c (- k^y \},$$
** hence $\phi(x,y)$ has a USHD over $k$.)
*e)*
Proofidea:
Compress $f^{-1}(C_0)$ with $\zeta(x,\~d)$,
then find $C_1 (= C_0$ bounded by the fibre size
such that if $f(\~d') = f(\~d)$ and $\zeta(x,\~d')$ implies the instances
for $C_1$, then it implies all.
Then ``exists such a $\~d'$ over $f(\~d)$'' is a USHD.
%
% More precisely:\
% ***) Let $a (- \M^x$ and $C_0 (=f C$.
% ** We have $\~d (- f^{-1}(C)$ such that $\tp(a/f^{-1}(C_0) -) \zeta(x,\~d) |- \tp_{\psi(x,f(z))}(a/f^{-1}(C_0))$,
% and $f(\~d) =: d (- C^{|z|}$.
% ** There is $C_1 (= C_0$ with $|C_1| <= |f^{-1}(d)|$,
% such that if $f(\~d') = d$ and
% $\zeta(x,\~d') |- \tp_{\psi(x,y)}(a/C_1)$,
% then already
% $\zeta(x,\~d') |- \tp_{\psi(x,y)}(a/C_0)$.
% ** Then $\exists \~d'. (f(\~d') = d /\
% [\zeta(x,\~d') |- \tp_{\psi(x,y))}(a/C_1)])$ isolates
% $\tp_{\psi(x,y)}(a/C_0)$ within $\tp(a/C_0)$.
% ** By the bound on $|C_1|$ this works uniformly.
% *e)*
.
**
Remarkn:
Following the proof gives a bound on the exponent in the distal cell
decompositions, hence in the incidence bound for $|x| = 1$, of
$t = 2(q^{d^3}+1)$ where $\res(k) = \F_q$.
For $|x| > 1$ a bound can in theory be computed, but it involves QE in
ACVF.
For the Szemerédi-Trotter case $\{ ((x,y),(a,b)) : y=ax+b \}$,
we get $t = 4(q+1)$, giving an exponent in the symmetric form of $\frac32
- \frac1{16(q+1)-2}$.
Question: in the Szemerédi-Trotter case with $\F_p(t)$, could there be an
exponent which doesn't depend on $p$? Worst lower bound I know is
$\frac43$ (by a similar "rectangular grid" argument as in the char 0
case).
.
*e)*
% One might wonder whether the use of Swiss cheese decomposition, i.e.
% canonical-up-to-finite boolean combinations, was really essential. I expect
% so. The following question formulates this more precisely; I expect the
% answer to be negative.
%Question:
% If $\phi(x,y)$ is a boolean combination of instances of $\psi(x,f_i(y))$
% where $f_i$ are algebraic quasifunctions,
% and $\psi$ has a USHD over $\bigcup_i f_i(C)$,
% does it follow that $\phi$ has a USHD over $C$?
% .
# Elekes-Szabó consequences
As in the characteristic 0 case, these incidence bounds yield ``modularity of
coherence'', and hence Elekes-Szabó bounds.
***) Let $k_0$ be a field admitting a valuation with finite residue field.
** For $r>=1$, let
$$k_r := \{ a (- (k_0)^\alg : \deg(k_0(a)/k_0) <= r \}.$$
** From the proof for $k_0$, we get that \underline{each $k_r$ is also distal}
in $(k_0)^\alg |= \ACVF$.
** This is sufficient for the arguments of one direction of the 1-dimensional case of the main result of B-Breuillard to go through:
Theoremn:
!renewtup a
!newop trd
!newop ccl
\providecommand{\bdl}{\boldsymbol\delta}
Let $\U$ be a non-principal ultrafilter on $\omega$.
Let $k' := ((k_0)^\U)^\alg <= ((k_0)^\alg)^\U =: L$,
so $k' = \bigcup_{r (- \omega} (k_r)^\U$.
Let $\xi (- \N^\U \\ \N$.
$$\bdl(\prod_{i --> \U} X_i) := \"st \log_\xi \lim_{i --> \U} |X_i| (- \R \cup \{\infty\}.$$
Equip $L$ with the structure generated by countably many internal
relations, including each $(k_r)^\U$, such that $\bdl$ is continuous.
For $\a (- L^{<\omega}$,
$$\bdl(\a) = \bdl(\tp(\a)) = \inf_{\phi (- \tp(\a)} \bdl(\phi(L)).$$
$P (= L$ is \defn{coherent} if $\bdl(\a) = \trd(\a)$ for all $\a (- P^{<\omega}$.
Write $\acl^0$ for field-theoretic algebraic closure.
Let $P (= k'$ be coherent, and let $\ccl(P) := \{a (- \acl^0(P) :
\bdl(a) = \trd(a) \} (= k'$.
Then $\ccl(P)$ is coherent, and $\acl^0|_{\ccl(P)}$ is a modular
pregeometry.
.
Using the Evans-Hrushovski characterisation of modular subpregeometries of
$\acl^0$ (proved by Evans-Hrushovski in arbitrary characteristic), one deduces:
Theoremn:
Let $k_0$ be a field admitting a valuation with finite residue field.
Let $V (= \A^m$ be an irreducible affine variety over $k_0$.
At least one of the following holds:
***(i) $\exists C,\eps > 0.\;\forall X_1,...,X_m (=f k_0.$
$$| V(k_0) \cap (X_1\times...\times X_m) | <= C\max(|X_i|)^{\dim V - \eps};$$
**(ii) Up to finite correspondences on co-ordinates and taking products,
\\\ $V$ is a subgroup of a power of a 1-dimensional algebraic group.
*ee*
.
%Remark:
% The higher dimensional ("coarse general position") version doesn't
% immediately go through, since it needs better bounds (a more uniform
% sublinearity in $s$ where we omit $K_{2,s}$, namely not depending on $k$
% where we omit $K_{k,k}$) and a positive characteristic version of higher
% dimensional Evans-Hrushovski. (The original ES case $m=3$ assuming general
% position (not just cgp) probably goes through directly, but I haven't
% checked.)
%
% Getting a formulation with "exactly one" would also take some more work.
% .
*e)*
\end{document}
~~