A New Criterion for Permutation Polynomials

Let v be the number of distinct values of a polynomial ƒ(x) of degree n over a finite field with q elements. Let u denote the degree of the first nonvanishing elementary symmetric function of the values of ƒ(x). Let w denote the degree of the first nonvanishing power sum of the values of ƒ(x). We study properties of u, w and relations between the quantities u, v, w, n, q. In particular, these investigations provide an easy proof of the recently discovered fact that v = q if v > q − (q − 1)/n. Many other characterizations of permutation polynomials are given and we prove several results concerning the number of elements c such that ƒ(x) + cx is a permutation polynomial. We explicitly evaluate u, v, w for n ≤ 3 and u, w for n = 4, p ≠ 2. The final section presents various examples and counterexamples.