Quotients of polynomial rings and regular t-balanced Cayley maps on abelian groups

TL;DR

This paper establishes a novel connection between polynomial ring quotients and regular t-balanced Cayley maps on abelian groups, leading to a comprehensive classification, especially for abelian 2-groups.

Contribution

It introduces a new approach linking polynomial rings to RBCM_t's and provides a complete classification for RBCM_t's on abelian 2-groups.

Findings

Established a connection between polynomial ring quotients and RBCM_t's.

Provided a complete classification of RBCM_t's on abelian 2-groups.

Developed a new method for classifying regular t-balanced Cayley maps.

Abstract

Given a finite group $\Gamma$, a regular $t$-balanced Cayley map (RBCM$_{t}$ for short) is a regular Cayley map $\mathcal{CM}(G,\Omega,\rho)$ such that $\rho(\omega)^{-1}=\rho^{t}(\omega)$ for all $\omega\in\Omega$. In this paper, we clarify a connection between quotients of polynomial rings and RBCM$_{t}$'s on abelian groups, so as to propose a new approach for classifying RBCM$_{t}$'s. We obtain many new results, in particular, a complete classification for RBCM$_{t}$'s on abelian 2-groups.