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

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.

On a paucity result in Incidence Geometry

“…where λ = 1 is a fixed number. If we consider a, c as variables and b, d as coefficients, then, clearly, (27) determines a family of conics and the number of the solutions to the equation can be estimated via the main result of [1], say.…”