Value Sets of Polynomials Over Finite Fields
Daqing Wan, Peter J.-S. Shiue, C. S. Chen · Proceedings of the American Mathematical Society · 1993
Let ${{\mathbf {F}}_q}$ be the finite field of $q$ elements, and let ${V_f}$ be the number of values taken by a polynomial $f(x)$ over ${{\mathbf {F}}_q}$. We establish a lower bound and an upper bound of ${V_f}$ in terms of certain invariants of $f(x)$. These bounds improve and generalize some of the previously known bounds of ${V_f}$. 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.