Reversed Dickson polynomials

TL;DR

This paper explores the permutation properties and fixed points of reversed Dickson polynomials over finite fields and integers, providing new insights and partial proofs related to their permutation behavior.

Contribution

It offers a detailed analysis of permutation and cycle structures of reversed Dickson polynomials and proves two special cases of a related conjecture.

Findings

Characterization of fixed points and cycle types over finite fields

Permutation behavior over inite rings and fields

Proof of two special cases of a permutation conjecture

Abstract

We investigate fixed points and cycle types of permutation polynomials and complete permutation polynomials arising from reversed Dickson polynomials of the first kind and second kind over $\mathbb{F}_p$. We also study the permutation behaviour of reversed Dickson polynomials of the first kind and second kind over $\mathbb{Z}_m$. Moreover, we prove two special cases of a conjecture on the permutation behaviour of reversed Dickson polynomials over $\mathbb{F}_p$.