Rédei permutations with the same cycle structure

Abstract

Let ${\mathbb{F}}_q$ be the finite field of order $q$, and ${\mathbb{P}}^1({\mathbb{F}}_q)={\mathbb{F}}_q\cup \{\infty \}$. Write $(x+\sqrt{y}{)}^m$ as $N(x,y)+D(x,y)\sqrt{y}$. For $m\in \mathbb{N}$ and $a\in {\mathbb{F}}_q$, the Rédei function $R_{m,a}:{\mathbb{P}}^1({\mathbb{F}}_q)\to {\mathbb{P}}^1({\mathbb{F}}_q)$ is defined by $N(x,a)/D(x,a)$ if $D(x,a)\ne 0$ and $x\ne \infty$, and $\infty$, otherwise. In this paper we give a complete characterization of all pairs $(m,n)\in {\mathbb{N}}^2$ such that the Rédei permutations $R_{m,a}$ and $R_{n,b}$ have the same cycle structure when $a$ and $b$ have the same quadratic character and $q$ is odd. We explore some relationships between such pairs $(m,n)$, and provide explicit families of Rédei permutations with the same cycle structure. When a Rédei permutation has a unique cycle structure that is not shared by any other Rédei permutation, we call it isolated. We show that the only isolated Rédei permutations are the isolated Rédei involutions. Moreover, all our results can be transferred to bijections of the form $mx$ and $x^m$ on certain domains.

References in corpus (1)

• Rédei permutations with cycles of the same length