Generalized cyclotomic mappings: Switching between polynomial, cyclotomic, and wreath product form

Abstract

This paper is concerned with so-called index $d$ generalized cyclotomic mappings of a finite field ${\mathbb{F}}_q$, which are functions ${\mathbb{F}}_q\to {\mathbb{F}}_q$ that agree with a suitable monomial function $x\mapsto a{x}^r$ on each coset of the index $d$ subgroup of ${\mathbb{F}}_q^{\ast }$. We discuss two important rewriting procedures in the context of generalized cyclotomic mappings and present applications thereof that concern index $d$ generalized cyclotomic permutations of ${\mathbb{F}}_q$ and pertain to cycle structures, the classification of $(q-1)$-cycles and involutions, as well as inversion.

60 pages