Complete mappings and Carlitz rank

TL;DR

This paper extends the understanding of complete mappings in finite fields by relating their existence to Carlitz rank, providing bounds and examples that generalize previous conjectures and results.

Contribution

It establishes new bounds on the Carlitz rank for the existence of complete mappings in finite fields and explores their properties and limitations.

Findings

No complete mappings of small Carlitz rank exist below half the field size.

Permutation polynomials with Carlitz rank at the threshold can be complete.

The results generalize the Chowla and Zassenhaus conjecture to broader settings.

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.