A divided-difference characterization of polynomials over finite fields of characteristic two

Abstract

For a finite field F of characteristic 2, an integer n≥ 3 , and for a function f: F→ F, if there is a function h: F→ F such that the divided difference f[x1, ⋯ , xn] on any n distinct elements of F satisfies f[x1, … , xn] = h(x1+ ⋯ + xn) , then f is a polynomial of degree at most n over F. This makes earlier work of Davies and Rousseau complete.