Dynamical Gröbner bases
type: Item
instance of: scholarly article
title: Dynamical Gröbner bases
zbMATH Open document ID: 1113.13013
author: Ihsen Yengui
published in: Journal of Algebra
publication date: 2006-12-07T00:00:00Z
review text: Since its introduction, the notion of Gröbner basis has been generalized in many ways. In [\textit{I. Yengui}, ``Computing a Gröbner basis of a polynomial ideal over a principal domain'', preprint (2004)], the author generalized Gröbner bases to discrete valuation domains, giving a simple decision procedure for the ideal membership problem in this case. The first part of the current paper is a direct generalization of that work to noetherian valuation rings. As a sample application, the author computes two examples of the structure of codes over two different finite-chain rings. A third example shows that in the integral case, the totally ordered group corresponding to the valuation must be archimedian in order to guarantee that Buchberger's Algorithm will terminate in a finite number of steps. The author then generalizes Gröbner bases to principal rings by computing a ``dynamical Gröbner basis'', i.e. a tree of Gröbner bases in localizations of the original ring at finitely generated multiplicative subsets. The author provides a detailed example of the construction, along with an example of how the ideal membership problem can be answered with the resulting basis. While the dynamical method avoids the ``expensive problem of factorizing an element in a principal domain into a finite product of irreducible elements'', it introduces its own computational problems in the form of redundant leaves ``due to the fact that one can obtain the same leaf in different ways''. The author does mention several shortcuts that may help to alleviate this problem.
Mathematics Subject Classification ID: 13P10
zbMATH DE Number: 5078115
zbMATH Keywords: dynamical Gröbner basis
zbMATH Keywords: ideal membership problem
zbMATH Keywords: valuation rings
zbMATH Keywords: noetherian rings
zbMATH Keywords: principal rings
reviewed by: Alyson Reeves
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2006.01.051
OpenAlex ID: W2081582540
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DOI: 10.1016/J.JALGEBRA.2006.01.051
description: scientific article; zbMATH DE number 5078115