Permutation binomials over finite fields

摘要

Let F-q 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 is an element of F-q for which the binomial x(n)(x(q-1/3) + 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. (c) 2021 Elsevier B.V. All rights reserved.