Expansions of the group $Z_8$ (Part I)
Miroslav Plo\v{s}\v{c}ica, Radka Schwartzov\'a, Ivana Varga

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 . This is the simplest open case of the problem, as the answer is known for (with prime) and, in general, reduces to the case when 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.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Polynomial and algebraic computation · Advanced Differential Equations and Dynamical Systems
