Reduced Gröbner Bases and Macaulay-Buchberger Basis Theorem over Noetherian Rings

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 Gröbner basis method of computing a $\mathbb{k}$-vector space basis of residue class polynomial rings over a field $\mathbb{k}$ (Macaulay-Buchberger Basis Theorem) to rings, i.e. $A[x_1,\dots ,x_n]/\mathfrak{a}$, where $\mathfrak{a}\subseteq A[x_1,\dots ,x_n]$ is an ideal. We give some insights into the characterization for two special cases, when $A=\mathbb{Z}$ and $A=\mathbb{k}[{\theta }_1,\dots ,{\theta }_m]$. 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.

References in corpus (1)

• The Groebner basis of the ideal of vanishing polynomials

Cited by in corpus (3)

• On Ideal Lattices, Gröbner Bases and Generalized Hash Functions
• Sandpile Groups of Cayley Graphs of ${\mathbb{F}}_2^r$
• On Gröbner Bases and Krull Dimension of Residue Class Rings of Polynomial Rings over Integral Domains