Value sets of sparse polynomials

TL;DR

This paper establishes a new uniform lower bound on the size of the value set of sparse polynomials over finite fields, improving upon previous bounds by incorporating arithmetic properties of the polynomial's degrees.

Contribution

It introduces a novel lower bound on the value set size of sparse polynomials that depends on the degrees and number of monomials, surpassing previous multiplicity-based bounds.

Findings

New lower bound on value set size for sparse polynomials

Bound depends on degrees and number of monomials

Stronger than previous bounds based on value multiplicities

Abstract

We obtain a new lower bound on the size of value set f(F_p) of a sparse polynomial f in F_p[X] over a finite field of p elements when p is prime. This bound is uniform with respect of 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 which canted be extracted from the bounds on multiplicities of individual values in f(F_p).