Reduced Gr\"obner Bases and Macaulay-Buchberger Basis Theorem over Noetherian Rings

TL;DR

This paper extends the theory of Gröbner bases and the Macaulay-Buchberger Basis Theorem from fields to Noetherian rings, enabling computation of bases in more general algebraic structures.

Contribution

It generalizes the characterization of residue class polynomial rings to Noetherian rings and extends border bases concepts to these rings, broadening computational algebra methods.

Findings

Extended Gröbner basis characterization to Noetherian rings.

Generalized Macaulay-Buchberger Basis Theorem for rings.

Applied border bases to rings with finitely generated free modules.

Abstract

In this paper, we extend the characterization of $\mathbb{Z}[x]/\ < f \ >$, where $f \in \mathbb{Z}[x]$ to be a free $\mathbb{Z}$-module to multivariate polynomial rings over any commutative Noetherian ring, $A$. The characterization allows us to extend the Gr\"obner basis method of computing a $\Bbbk$-vector space basis of residue class polynomial rings over a field $\Bbbk$ (Macaulay-Buchberger Basis Theorem) to rings, i.e. $A[x_1,\ldots,x_n]/\mathfrak{a}$, where $\mathfrak{a} \subseteq A[x_1,\ldots,x_n]$ is an ideal. We give some insights into the characterization for two special cases, when $A = \mathbb{Z}$ and $A = \Bbbk[\theta_1,\ldots,\theta_m]$. As an application of this characterization, we show that the concept of border bases can be extended to rings when the corresponding residue class ring is a finitely generated, free $A$-module.