Generalized Cyclotomic Mappings: Switching Between Polynomial, Cyclotomic, and Wreath Product Form

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