Applications of the Hasse–Weil bound to permutation polynomials

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 F q where q is sufficiently large. To facilitate these applications, the absolute irreducibility of certain polynomials in F q [ X , Y ] is established.