On the Carlitz Rank of Permutation Polynomials Over Finite Fields: Recent Developments

Abstract

The Carlitz rank of a permutation polynomial over a finite field Fq is a simple concept that was introduced in the last decade. In this survey article, we present various interesting results obtained by the use of this notion in the last few years. We emphasize the recent work of the authors on the permutation behavior of polynomials f + g, where f is a permutation over Fq of a given Carlitz rank, and g∈Fq[x] is of prescribed degree. The relation of this problem to the well-known Chowla–Zassenhaus conjecture is described. We also present some initial observations on the iterations of a permutation polynomial f∈Fq[x] and hence on the order of f as an element of the symmetric group S q .