Local permutation polynomials and their companions

TL;DR

This paper extends the study of local permutation polynomials over finite fields by introducing new families in even characteristic and analyzing the existence of companions, revealing cases where companions do or do not exist.

Contribution

It introduces three new families of local permutation polynomials over finite fields of even characteristic and investigates the existence of their companions.

Findings

0-Klenian polynomials in even characteristic have no companions

Explicit companions are provided for e ≥ 1 in even characteristic

New families of local permutation polynomials are constructed

Abstract

Gutierrez and Urroz (2023) have proposed a family of local permutation polynomials over finite fields of arbitrary characteristic based on a class of symmetric subgroups without fixed points called $e$-Klenian groups. The polynomials within this family are referred to as $e$-Klenian polynomials. Furthermore, they have shown the existence of companions for the $e$-Klenian polynomials when the characteristic of the finite field is odd. Here, we present three new families of local permutation polynomials over finite fields of even characteristic. We also consider the problem of the existence of companions for the $e$-Klenian polynomials over finite fields of even characteristic. More precisely, we prove that over finite fields of even characteristic, the $0$-Klenian polynomials do not have any companions. However, for $e \geq 1$, we explicitly provide a companion for the $e$-Klenian polynomials. Moreover, we provide a companion for each of the new families of local permutation polynomials that we introduce.