Strong Gr\"obner bases and linear algebra in multivariate polynomial rings over Euclidean domains

TL;DR

This paper introduces methods for computing Gr"obner bases in multivariate polynomial rings over Euclidean domains and demonstrates their application to solving linear systems in these rings.

Contribution

It provides a self-contained introduction to Gr"obner bases over Euclidean domains and explains their use in solving linear systems in multivariate polynomial rings.

Findings

Effective algorithms for Gr"obner bases over Euclidean domains.

Application of these bases to solve linear systems.

Comprehensive, self-contained exposition.

Abstract

We provide a self-contained introduction to Gr\"obner bases of submodules of $R[x_1, \ldots, x_n]^k$, where $R$ is a Euclidean domain, and explain how to use these bases to solve linear systems over $R[x_1, \ldots, x_n]$.