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

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.

Categories: math.CO