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 Fq, which are functions Fq → Fq that agree with a suitable monomial function x 7→ axr on each coset of the index d subgroup of Fq. 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 Fq and pertain to cycle structures, the classification of (q − 1)-cycles and involutions, as well as inversion.