On Gr\"obner bases over Dedekind domains

TL;DR

This paper extends the theory and algorithms of Gr"obner bases from fields and principal ideal rings to polynomial rings over Dedekind domains, broadening their applicability in algebraic computations.

Contribution

It generalizes the concept of Gr"obner bases to Dedekind domains using finitely generated projective modules, providing both theoretical insights and algorithmic approaches.

Findings

Describes Gr"obner bases over Dedekind domains.

Provides a theoretical framework similar to PIDs.

Develops algorithms for computing Gr"obner bases in this setting.

Abstract

Gr\"obner bases are a fundamental tool when studying ideals in multivariate polynomial rings. More recently there has been a growing interest in transferring techniques from the field case to other coefficient rings, most notably Euclidean domains and principal ideal rings. In this paper we will consider multivariate polynomial rings over Dedekind domain. By generalizing methods from the theory of finitely generated projective modules, we show that it is possible to describe Gr\"obner bases over Dedekind domains in a way similar to the case of principal ideal domains, both from a theoretical and algorithmic point of view.