A characterization of stable and Borel ideals
type: Item
instance of: scholarly article
title: A characterization of stable and Borel ideals
zbMATH Open document ID: 1120.13018
author: Maria Grazia Marinari
author: Luciana Ramella
published in: Applicable Algebra in Engineering, Communication and Computing
publication date: 2006-02-08T00:00:00Z
review text: The authors study special classes of monomial ideals in the polynomial ring over a field of characteristic zero. For every monomial ideal \(a\) they consider the order ideal \({\mathcal N}(a)\) associated to \(a\), that is, the set of the terms outside \(a\). The main result of the paper is a characterization of stable and Borel-fixed ideals in terms of the corresponding order ideals. Moreover the authors associate to every monomial ideal \(a\) the so-called \(\varepsilon\)-vectors, denoted by \(\varepsilon(a)\), consisting of a sequence of symbols \(\infty\), positive integers and finite sets of /'s. Such a vector provides strong and compact information on the ``geometric configuration'' of the order ideal \({\mathcal N}(a)\) and on the generators of \(a\). In particular, if \(a\) is stable then the knowledge of \(\varepsilon(a)\) allows to write the minimal system of generators of \(a\). In the special case of lex-segment ideals a full characterization of the possible \(\varepsilon\)-vectors is given.
reviewed by: Emanuela De Negri
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13C13
zbMATH DE Number: 5004483
zbMATH Keywords: monomial ideals
zbMATH Keywords: stable ideals
zbMATH Keywords: Borel-fixed ideals
zbMATH Keywords: lex-segment
MaRDI profile type: Publication
full work available at URL: s00200-005-0171-2
OpenAlex ID: W2046655238
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DOI: 10.1007/S00200-005-0171-2
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description: scientific article; zbMATH DE number 5004483