Calcul matriciel g\'en\'eralis\'e sur les domaines de Pr\"ufer

TL;DR

This paper introduces an algorithm for computing the Hermite normal form of pseudo-matrices over Pr"ufer domains, generalizing existing methods and providing new insights into modules and linear systems over these domains.

Contribution

It presents a novel algorithm for Hermite normal form over Pr"ufer domains, extending Cohen's methodology for Dedekind domains and exploring Smith normal form in dimension one.

Findings

Algorithm successfully computes Hermite normal form over Pr"ufer domains.

Constructive proofs of key theoretical results on finitely presented modules.

Results on Smith normal form in Pr"ufer domains of dimension one.

Abstract

In this paper, we first present an algorithm for computing the Hermite normal form of pseudo-matrices over Pr\"ufer domains. This algorithm allows us to provide constructive proofs of the main theoretical results on finitely presented modules over Pr\"ufer domains and to discuss the resolution of linear systems. In some sense, we generalize the methodology developed by Henri Cohen for Dedekind domains. Finally, we present some results over Pr\"ufer domains of dimension one about the Smith normal form.