On the Carlitz rank of permutation polynomials

A well-known result of Carlitz, that any permutation polynomial p(x) of a finite field F-q is a composition of linear polynomials and the monomial x(q-2). implies that V(x) can be represented by a polynomial P-n(x) = (...((a(0)x + a(1))(q-2) + a(2))(q-2)...+ a(n))(q-2) + a(n+1). for some n >= 0. The smallest integer n, such that P,,(x) represents p(x) is of interest since it is the least number of "inversions" x(q-2), needed to obtain p(x). We define the Carlitzrank of p(x) as n, and focus here on the problem of evaluating it. We also obtain results on the enumeration of permutations of F-q with a fixed Carlitz rank

On the carlitz rank of permutation polynomials over finite fields:recent developments

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