Computing ideals of points
type: Item
instance of: scholarly article
title: Computing ideals of points
zbMATH Open document ID: 0977.13011
author: John Abbott
author: Anna M. Bigatti
author: Martin Kreuzer
author: Lorenzo Robbiano
published in: Journal of Symbolic Computation
publication date: 2001-03-11T00:00:00Z
review text: \textit{H. M. Möller} and \textit{B. Buchberger} [Computer algebra, EUROCAM '82, Conf. Marseille/France 1982, Lect. Notes Comput. Sci. 144, 24-31 (1982; Zbl 0549.68026)] gave an algorithm to compute generators of the defining ideal of a finite set of points in affine space. While this algorithm is very efficient over finite fields, one may run into the problem of very large coefficients when performing this algorithm over the rationals. The first main result in the article under review is a modular version of the Buchberger-Möller algorithm to deal with this difficulty. Next, the authors consider the problem of determining generators for the homogeneous ideal of a finite set of points in projective space. This has been considered previously by \textit{M. G. Marinari, H. M. Möller}, and \textit{T. Mora} [Appl. Algebra Eng. Commun. Comput. 4, No. 2, 103-145 (1993; Zbl 0785.13009)], and by \textit{F. Cioffi} [Ric. Mat. 48, No. 1, 55-63 (1999; Zbl 0982.13015)]. For this problem, a stopping criterion is a difficulty, since the ideal is now one-dimensional instead of zero-dimensional. The authors show that a lemma (lemma 12.1) in the cited article by Marinari et al. is incorrect, and then go on to use properties of the Hilbert function to derive a stopping criterion. Finally, the authors consider the complexity of their algorithms and give examples of their algorithms implemented in CoCoA.
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 14Q20
Mathematics Subject Classification ID: 14M05
zbMATH DE Number: 1538748
zbMATH Keywords: Gröbner basis
zbMATH Keywords: Buchberger-Möller algorithm
zbMATH Keywords: Hilbert function
zbMATH Keywords: finite set of points
zbMATH Keywords: algorithm over the rationals
zbMATH Keywords: CoCoA
reviewed by: Robert F. Lax
describes a project that uses: CoCoA
MaRDI profile type: Publication
full work available at URL: jsco.2000.0411
OpenAlex ID: W1982662469
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DOI: 10.1006/JSCO.2000.0411
description: scientific article; zbMATH DE number 1538748