Value Sets of Sparse Polynomials

Abstract

AbstractWe obtain a new lower bound on the size of the value set $\mathscr{V}(f)=f(\mathbb{F}_{p})$ of a sparse polynomial $f\in \mathbb{F}_{p}[X]$ over a finite field of $p$ elements when $p$ is prime. This bound is uniform with respect to the degree and depends on some natural arithmetic properties of the degrees of the monomial terms of $f$ and the number of these terms. Our result is stronger than those that can be extracted from the bounds on multiplicities of individual values in $\mathscr{V}(f)$.