Redei permutations with the same cycle structure

Abstract: Permutation polynomials over finite fields have been extensively studied over the past decades. Among the major challenges in this area are the questions concerning their cycle structures as they capture relevant properties, both theoretically and practically. In this talk we focus on a family of permutation polynomials, the so called R\'edei permutations. Although their cycle structures are known, there are other related questions that can be investigated. For example, when do two R\'edei permutations have the same cycle structure? We give a characterization of such pairs, and present explicit families of R\'edei permutations with the same cycle structure. We also discuss some results regarding R\'edei permutations with a particularly simple cycle structure, consisting of $1$- and $j$-cycles only, when $j$ is $4$ or a prime number. The case $j = 2$ is specially important in some applications. We completely describe R\'edei involutions with a prescribed cycle structure, and show that the only R\'edei permutations with a unique cycle structure are the involutions.

This is joint work with Juliane Capaverde and Virg\'inia Rodrigues.

number theory

Audience: researchers in the discipline

Combinatorial and additive number theory (CANT 2022)