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

Given a finite group $\mathrm{\Gamma }$, a regular $t$-balanced Cayley map (RBCM${}_t$ for short) is a regular Cayley map $\mathcal{C}\mathcal{M}(G,\mathrm{\Omega },\rho )$ such that $\rho (\omega {)}^{-1}={\rho }^t(\omega )$ for all $\omega \in \mathrm{\Omega }$. 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.