Incidence geometry and polynomial expansion over finite fields

TL;DR

This paper develops a higher-degree analogue of a Szemerédi–Trotter theorem over finite fields using spectral theory and algebraic geometry, with applications to polynomial expansion.

Contribution

It introduces a novel higher-degree incidence bound over finite fields, extending classical geometric combinatorics results.

Findings

Established a new incidence bound for higher-degree curves over finite fields

Demonstrated applications to polynomial expansion problems

Extended spectral and algebraic methods to finite field geometry

Abstract

We use spectral theory and algebraic geometry to establish a higher-degree analogue of a Szemer\'edi--Trotter-type theorem over finite fields, with an application to polynomial expansion.