Szemer\'{e}di-Trotter type results in arbitrary finite fields

TL;DR

This paper establishes explicit upper bounds on incidences between lines and Cartesian products in finite fields and applies these results to derive new sum-product estimates involving reciprocals.

Contribution

It provides novel explicit incidence bounds in finite fields and leverages these to improve sum-product estimates involving reciprocals.

Findings

Established explicit upper bounds on line-point incidences in finite fields.

Derived new sum-product estimates involving sums of reciprocals.

Enhanced understanding of combinatorial structures in finite fields.

Abstract

Let $q$ be a power of a prime and $\mathbb{F}_q$ the finite field consisting of $q$ elements. We prove explicit upper bounds on the number of incidences between lines and Cartesian products in $\mathbb{F}_q^2$. We also use our results on point-line incidences to give new sum-product type estimates concerning sums of reciprocals.