Permutation binomials over finite fields

Abstract

Let Fq denote the finite field with q elements. In this paper we use the relation between suitable polynomials and number of rational points on algebraic curves to give the exact number of elements a∈Fq for which the binomial xn(xq−13+a) is a permutation polynomial. In order to do this, we employ results on the Cartin-Manin operator and the Riemann Hypothesis for elliptic curves to present the exact number of points on a suitable elliptic curve.