Computing the \textbf{V}-saturation of finitely-generated submodules of \(\mathbf{V} [X]^m\) where \textbf{V} is a valuation domain
type: Item
instance of: scholarly article
title: Computing the \textbf{V}-saturation of finitely-generated submodules of \(\mathbf{V} [X]^m\) where \textbf{V} is a valuation domain
zbMATH Open document ID: 1327.13073
author: Lionel Ducos
author: Samiha Monceur
author: Ihsen Yengui
published in: Journal of Symbolic Computation
publication date: 2015-08-19T00:00:00Z
review text: The article describes an algorithm to compute the \(\mathbf{V}\)-saturation of a finitely generated submodule of \(\mathbf{V}[X]^n\), where \(V\) is a saturation domain, breaking it down into several specific subalgorithms and analyzing its complexity. Here, \(X\) denotes only one variable, so the algorithm is for the univariate case; the authors nod to a generalization for the multivariate case of Krull dimension 1; this result applies to the univariate case of arbitrary Krull dimension. The article proves (a) correctness if the algorithm terminates; (b) termination after a specified number of operations; and (c) complexity bounds. Two applications are given: a test for an analogue of ideal membership called saturation membership, and the computation of syzygies over the polynomial ring of a valuation domain, proving the special case that in the case of a Prüfer domain \(\mathbf{R}\) the \(\mathbf{R}\)-saturation of a finitely-generated submodule of \(\mathbf{R}[X]^m\) is finitely generated and the polynomial ring \(\mathbf{R}[X]\) is coherent.
reviewed by: John Perry
Mathematics Subject Classification ID: 13F30
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 13P99
Mathematics Subject Classification ID: 68Q25
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 68W40
zbMATH DE Number: 6473752
zbMATH Keywords: saturation
zbMATH Keywords: coherence
zbMATH Keywords: echelon matrix
zbMATH Keywords: syzygies
zbMATH Keywords: valuation domains
zbMATH Keywords: Prüfer domains
MaRDI profile type: Publication
full work available at URL: j.jsc.2015.02.006
OpenAlex ID: W2031459264
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DOI: 10.1016/J.JSC.2015.02.006
description: scientific article; zbMATH DE number 6473752