On permutation polynomials over finite fields: differences and iterations

The Carlitz rank of a permutation polynomial f over a finite field Fq is a simple concept that was introduced in the last decade. Classifying permutations over Fq
with respect to their Carlitz ranks has some advantages, for instance f with a given Carlitz rank can be approximated by a rational linear transformation. In this note we present our recent results on the permutation behaviour of polynomials f+g, where f is a permutation over Fq of a given Carlitz rank, and g ∈ Fq[x] is of prescribed degree. We describe the relation of this problem to the well-known Chowla-Zassenhaus conjecture. We also study iterations of permutation polynomials by using the approximation property that is mentioned above.