Applications of the Hasse–Weil bound to permutation polynomials

Abstract

Riemann's hypothesis on function fields over a finite field implies the Hasse–Weil bound for the number of zeros of an absolutely irreducible bi-variate polynomial over a finite field. The Hasse–Weil bound has extensive applications in the arithmetic of finite fields. In this paper, we use the Hasse–Weil bound to prove two results on permutation polynomials over Fq where q is sufficiently large. To facilitate these applications, the absolute irreducibility of certain polynomials in Fq[X,Y] is established.