Advances and improvements in the theory of standard bases and syzygies
type: Item
instance of: scholarly article
title: Advances and improvements in the theory of standard bases and syzygies
zbMATH Open document ID: 0854.13015
DOI: 10.1007/BF01273348
author: Gert-Martin Greuel
author: Gerhard Pfister
published in: Archiv der Mathematik
publication date: 1996-11-22T00:00:00Z
review text: The aim of this article is to describe recent advances and improvements on the tangent cone algorithm of \textit{T. Mora}. This tangent cone algorithm is itself a variant of B. Buchberger's celebrated algorithm for constructing a Gröbner basis of an ideal in a polynomial ring over a field. In the same manner as the knowledge of a Gröbner basis allows the computation of numerous invariants of the coordinate ring of a projective algebraic variety, a standard basis (computed by the tangent cone algorithm) does so for invariants of the local ring of an algebraic variety at a given point. In this paper we describe a generalization which includes Buchberger's and Mora's algorithm as special cases. That is, we prove -- with an appropriate definition of ecart -- that Mora's algorithm terminates for any ordering on the monomials of \(K[x_1, \dots, x_n]\), which is compatible with the natural semigroup structure (a fact which was found independently by Gräbe), in particular, the variables may have as well negative, positive or zero weights. More or less all algorithms using Gröbner bases (such as computation of syzygies, ideal theoretic operations, etc.) are now available in this general context. Our generalization provides also an easy manner to implement standard bases for modules over the Weyl algebra and for \({\mathcal D}\)-modules. The authors give also a proof of (a sharpened version) of Hilbert's syzygy theorem for localizations of polynomial rings over a field with respect to a general semigroup ordering. This proof, which generalizes Schreyer's proof for polynomial rings, uses just the existence of standard bases and is quite elementary. Using a deformation connecting the module of leading terms with the module itself, it is proven that standard bases can be used to compute numerical invariants of a module like the dimension, and under certain natural restrictions on the ordering in the local case, the multiplicity and Hilbert function. Computer experiments with an …
reviewed by: Gerhard Pfister
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 14Q99
zbMATH DE Number: 937062
zbMATH Keywords: computer algebra system SINGULAR
zbMATH Keywords: tangent cone algorithm
zbMATH Keywords: Gröbner basis
zbMATH Keywords: ideal in a polynomial ring
zbMATH Keywords: standard basis
zbMATH Keywords: invariants of the local ring of an algebraic variety
zbMATH Keywords: Hilbert's syzygy theorem
zbMATH Keywords: deformation
zbMATH Keywords: dimension
zbMATH Keywords: multiplicity
zbMATH Keywords: Hilbert function
describes a project that uses: SINGULAR
describes a project that uses: CoCoA
describes a project that uses: Macaulay2
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01273348
OpenAlex ID: W2056117826
description: scientific article; zbMATH DE number 937062