Relative distality and incidence bounds in positive characteristic

Abstract

We prove Szemerédi-Trotter-type incidence bounds for algebraic relations in any field $F$ admitting a valuation with finite residue field. This includes $\mathbb F_q(t)$, $\mathbb F_q((t))$, and all finitely generated extensions of finite fields. Bays and Martin show that any algebraic $K_{d,s}$-free relation $E\subseteq F^k\times F^\ell$ admits a power-saving in the classical Zarankiewicz bound of Kővári, Sós, and Turán. Answering a question of the authors, we show that this power-saving is uniform in $F$: There is some $ε>0$ depending only on $d$ and the complexity of $E$ (and not on $F$) such that for $A\subseteq F^k$ and $B\subseteq F^\ell$, the number of edges $|E(A,B)|$ is at most $O(|A|^{1-ε}|B|^{1-\frac{1-ε}{d}}+|A|+|B|)$. For the case where $A$ is a set of points and $B$ is a set of lines in $F^2$ and $E$ is the incidence relation, we prove the explicit bound $O(|P|^{\frac{8}{9}}|L|^{\frac{5}{9}}+|P|+|L|)$. As the central ingredient of the proof, we introduce the model-theoretic notion of distality relative to a definable set. We begin developing this notion and prove that the theory of algebraically closed valued fields is distal relative to the residue field. An application of a result of Chernikov, Peterzil, and Starchenko, based on work of Chernikov, Galvin, and Starchenko, yields our incidence bounds.