Expansions of the group $Z_8$ (Part I)

TL;DR

This paper explores the complex structure of polynomial clones over the cyclic group Z_8, focusing on those with even coefficients in nonlinear monomials, marking progress in understanding polynomial clones over composite groups.

Contribution

It provides a partial classification of polynomial clones over Z_8 with even coefficients, addressing a previously open case in the study of polynomial clones over finite cyclic groups.

Findings

Partial description of polynomial clones over Z_8

Focus on polynomials with even nonlinear coefficients

Highlights complexity of the Z_8 case

Abstract

We investigate clones in the interval between the group polynomials and the ring polynomials of ${\mathbb Z}_8$. This is the simplest open case of the problem, as the answer is known for ${\mathbb Z}_{p^2}$ (with $p$ prime) and, in general, ${\mathbb Z}_n$ reduces to the case when $n$ is a prime power. The investigated structure proves to be very complicated, so we provide only a partial description. We restrict our attention to polynomials whose nonlinear monomials have even coefficients.