Value sets of polynomials over finite fields

Abstract

Let F q {{\mathbf {F}}_q} be the finite field of q q elements, and let V f {V_f} be the number of values taken by a polynomial f ( x ) f(x) over F q {{\mathbf {F}}_q} . We establish a lower bound and an upper bound of V f {V_f} in terms of certain invariants of f ( x ) f(x) . These bounds improve and generalize some of the previously known bounds of V f {V_f} . In particular, the classical Hermite-Dickson criterion is improved. Our bounds also give a new proof of a recent theorem of Evans, Greene, and Niederreiter. Finally, we give some examples which show that our bounds are sharp.