Canonical Number Systems on Polynomial Quotients: A Finite-Generators Pipeline

I introduce an n-dimensional canonical number system (CNS) on modules: a finite base frame b, an E-linear shift T, and a finite digit set D yield unique finite expansions along places \( \{T^{k}(b_i)\} \). For a finitely generated ring \( R=\mathbb Z[a_1,\dots,a_n] \), I use the presentation \( R\cong \mathbb Z[x_1,\dots,x_n]/\ker(\phi) \) and compute \( \ker(\phi) \) via a graph-ideal elimination scheme. When the quotient has finitely many standard monomials, choosing a non–zero-divisor prime p equips R with a base-p CNS (pre-folding) that uses those monomials as places. A digit-folding lemma then compresses coordinates whenever some place has only finitely many powers, preserving uniqueness. This provides a constructive pipeline from presentations of finitely generated, countable rings to explicit multi-dimensional CNS representations, supporting the conjecture that every such ring is isomorphic to an n-dimensional CNS.