Incidence bounds in positive characteristic via valuations and distality

TL;DR

This paper establishes incidence bounds in positive characteristic fields using valuations and distality, leading to Szemerédi-Trotter-like results and a version of the Elekes-Szabó theorem.

Contribution

It introduces a novel approach connecting distality and valuations to derive incidence bounds in positive characteristic fields.

Findings

Proves distality of quantifier-free relations on valued fields with finite residue field.

Derives Szemerédi-Trotter-like incidence bounds for function fields over finite fields.

Establishes a version of the Elekes-Szabó theorem in this setting.

Abstract

We prove distality of quantifier-free relations on valued fields with finite residue field. By a result of Chernikov-Galvin-Starchenko, this yields Szemer\'edi-Trotter-like incidence bounds for function fields over finite fields. We deduce a version of the Elekes-Szab\'o theorem for such fields.