Sparsity of curves and additive and multiplicative expansion of rational maps over finite fields

TL;DR

This paper establishes bounds on solutions to polynomial equations over finite fields within small sets, and applies these results to show that certain sequences generated by rational functions do not cluster in small intervals or subgroups.

Contribution

It provides new upper bounds on solutions to polynomial equations over finite fields in small sets and applies these bounds to properties of sequences generated by rational functions.

Findings

Bounds on solutions to polynomial equations in small sets over finite fields.

Sequences from rational functions do not cluster in small intervals or subgroups.

Analogue of Bombieri-Pila results in positive characteristic.

Abstract

For a prime $p$ and a polynomial $F(X,Y)$ over a finite field $\mathbb{F}_p$ of $p$ elements, we give upper bounds on the number of solutions $$ F(x,y)=0, \quad x\in\mathcal{A}, \ y\in \mathcal{B}, $$ where $\mathcal{A}$ and $\mathcal{B}$ are very small intervals or subgroups. These bounds can be considered as positive characteristic analogues of the result of Bombieri and Pila (1989) on sparsity of integral points on curves. As an application we prove that distinct consecutive elements in sequences generated compositions of several rational functions are not contained in any short intervals or small subgroups.