Complete mappings and Carlitz rank

Abstract

The well-known Chowla and Zassenhaus conjecture, proven by Cohen in 1990, states that for any $d\ge 2$ and any prime $p>(d^2-3d+4)^2$ there is no complete mapping polynomial in $\mathbb {F}_p[x]$ of degree d. For arbitrary finite fields $\mathbb {F}_q$, we give a similar result in terms of the Carlitz rank of a permutation polynomial rather than its degree. We prove that if $n<\lfloor q/2\rfloor$, then there is no complete mapping in $\mathbb {F}_q[x]$ of Carlitz rank n of small linearity. We also determine how far permutation polynomials f of Carlitz rank $n<\lfloor q/2\rfloor$ are from being complete, by studying value sets of $f+x.$ We provide examples of complete mappings if $n=\lfloor q/2\rfloor$, which shows that the above bound cannot be improved in general.