Generalized Cyclotomic Mappings: Switching Between Polynomial, Cyclotomic, and Wreath Product Form (Communications in Mathematical Research, 2022)
This paper is concerned with so-called index $d$ generalized cyclotomic mappings of a finite field $\mathbb{F}_q$, which are functions $\mathbb{F}_q → \mathbb{F}_q$ that agree witha suitable monomial function $x\mapsto ax^r$ on each coset of the index $d$ subgroup of $\mathbb{F}^∗_q$. 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.
Keywords: Finite fields; cyclotomy; cyclotomic mappings; permutation polynomials; wreath product; cycle structure; involution.