On permutation binomials over finite fields

Abstract

Let Fq be the finite field of characteristic p containing q= pr elements and f(x)= axn + xm , a binomial with coefficients in this field. If some conditions on the greatest common divisor of n- m and q- 1 are satisfied then this polynomial does not permute the elements of the field. We prove in particular that if f(x)= axn + xm permutes Fq , where n> m> 0 and a Fq , then p- 1≤ (d- 1)d, where d= gcd (n- m, p- 1), and that this bound of p, in terms of d only, is sharp. We show as well how to obtain in certain cases a permutation binomial over a subfield of Fq from a permutation binomial over Fq . © 2013 Australian Mathematical Publishing Association Inc.