A modular Szemeredi-Trotter theorem for hyperbolas

TL;DR

This paper extends the Szemeredi-Trotter theorem to hyperbolas over finite fields, providing new combinatorial bounds for incidences between points and hyperbolic curves.

Contribution

It introduces a modular version of the Szemeredi-Trotter theorem specifically for hyperbolas in finite fields, a novel extension of classical incidence geometry results.

Findings

Established incidence bounds for hyperbolas over finite fields

Extended combinatorial geometry techniques to new algebraic curves

Provided a framework for future incidence problems in finite fields

Abstract

We establish a Szemeredi-Trotter type result for hyperbolas in $\mathbb F_p\times\mathbb F_p$