Characterizations of border bases
type: Item
instance of: scholarly article
title: Characterizations of border bases
zbMATH Open document ID: 1081.13011
author: Achim Kehrein
author: Martin Kreuzer
published in: Journal of Pure and Applied Algebra
publication date: 2005-04-07T00:00:00Z
review text: Border bases are a system of generators of some zero-dimensional ideals which behave numerically better than Gröbner bases. The usual presentations and characterizations of border bases differ from those of Gröbner bases. In this paper border bases are presented and characterized as Gröbner bases. There is a border division algorithm in which degree is replaced by border index. Then the border bases are characterized via the special generation property. Normal forms and reduction process are also developed (however reduction is not Noetherian). In order to introduce a Buchberger criterion the authors characterize the border bases in terms of the commutation of the multiplication matrices (the ideals are zero dimensional); then they define neighborhood between border terms and these are used to prove the Buchberger criterion for border bases. Finally the paper covers the computation of syzygies; the authors compute syzygies for border terms and these are lifted to border bases; once again the neighbor borders make the computation of syzygies easier.
reviewed by: F. J. Lobillo
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 65H10
zbMATH DE Number: 2153395
zbMATH Keywords: border bases
zbMATH Keywords: system of generators
zbMATH Keywords: zero-dimensional ideals
zbMATH Keywords: Gröbner bases
zbMATH Keywords: border division algorithm
zbMATH Keywords: border index
zbMATH Keywords: special generation property
zbMATH Keywords: syzygies
MaRDI profile type: Publication
full work available at URL: j.jpaa.2004.08.028
OpenAlex ID: W2099652582
cites work: Q3806684
cites work: Families of estimable terms
cites work: Q5505187
cites work: Q4507801
cites work: Q4279505
cites work: Q4502650
cites work: Numerical Polynomial Algebra
DOI: 10.1016/J.JPAA.2004.08.028
description: scientific article; zbMATH DE number 2153395