On an implementation of standard bases and syzygies in SINGULAR
type: Item
instance of: scholarly article
title: On an implementation of standard bases and syzygies in SINGULAR
zbMATH Open document ID: 0854.13016
DOI: 10.1007/BF01190332
author: Gert-Martin Greuel
author: Bernd Martin
author: W. Neumann
author: Gerhard Pfister
author: W. Pohl
author: Hans Schönemann
author: Hubert Grassmann
author: Thomas Siebert
published in: Applicable Algebra in Engineering, Communication and Computing
publication date: 1996-06-23T00:00:00Z
review text: In a previous article [Arch. Math. 66, No. 2, 163-176 (1996; see the preceding review)], the second and fifth authors described a generalization of the algorithms of Buchberger and Mora to the case of any ordering on the monomials in a polynomial ring that is compatible with the semigroup structure. (The variables may have negative, positive, or zero weights.) In the present article, the authors describe the implementation of these ideas in the system SINGULAR. In the first part of the paper, the implementation of the standard basis algorithm is presented. Two methods that can improve the efficiency of this algorithm are the HCtest and ecartMethod. The HCtest is applicable only to 0-dimensional ideals and involves computing the minimal monomial not contained in the initial ideal (the highest corner) and discarding all bigger monomials in further computations. The ecartMethod is applicable in all cases and provides an automatic choice of a weight vector of positive integers such that the weight ecart of the input polynomials become as small as possible. Results of twenty standard basis computations with various options (e.g. Lazard's method, Mora's algorithm with sugar, HCtest, ecartMethod) are given. In the last part of the paper, the authors compare different methods for computing syzygies and minimal resolutions in twenty cases. Included in the first implementation of F.-O. Schreyer's algorithm for computing syzygies. No single strategy is best in all cases.
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 68W30
zbMATH DE Number: 878206
zbMATH Keywords: system SINGULAR
zbMATH Keywords: standard basis algorithm
zbMATH Keywords: HCtest
zbMATH Keywords: syzygies
zbMATH Keywords: minimal resolutions
reviewed by: Robert F. Lax
describes a project that uses: CoCoA
describes a project that uses: Macaulay2
describes a project that uses: SINGULAR
MaRDI profile type: Publication
cites work: A criterion for detecting m-regularity
cites work: Q4237372
cites work: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal
cites work: Q3714165
cites work: Q4317713
cites work: The tangent cone algorithm and homogenization
cites work: On an installation of Buchberger's algorithm
cites work: Q4237370
cites work: Advances and improvements in the theory of standard bases and syzygies
cites work: Q3685189
cites work: Q3804185
cites work: Q3729028
cites work: New constructive methods in classical ideal theory
cites work: Q4234258
cites work: A standard basis approach to syzygies of canonical curves.
full work available at URL: https://doi.org/10.1007/bf01190332
OpenAlex ID: W2044449640
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description: scientific article; zbMATH DE number 878206