A Point-Conic Incidence Bound and Applications over $\mathbb F_p$

TL;DR

This paper establishes the first incidence bound between points and conics over prime fields and applies it to improve results in polynomial expansion, distance problems, and a variant of Beck's theorem in finite field geometry.

Contribution

It introduces a novel incidence bound for points and conics over prime fields and applies it to various problems in finite field combinatorics and geometry.

Findings

New lower bounds on algebraic distances

Improved size estimates for distance sets

A variant of Beck's theorem for conics

Abstract

In this paper, we prove the first incidence bound for points and conics over prime fields. As applications, we prove new results on expansion of bivariate polynomial images and on certain variations of distinct distances problems. These include new lower bounds on the number of pinned algebraic distances as well as improvements of results of Koh and Sun (2014) and Shparlinski (2006) on the size of the distance set formed by two large subsets of finite dimensional vector spaces over finite fields. We also prove a variant of Beck's theorem for conics.