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

Given a finite group Γ, a regular t-balanced Cayley map (RBCMt for short) is a regular Cayley map CM(G,Ω,ρ) such that ρ(ω)−1=ρt(ω) for all ω∈Ω. In this paper, we clarify a connection between quotients of polynomial rings and RBCMt's on abelian groups, so as to propose a new approach for classifying RBCMt's. We obtain many new results, in particular, a complete classification for RBCMt's on abelian 2-groups.