Noether normalization guided by monomial cone decompositions
type: Item
instance of: scholarly article
title: Noether normalization guided by monomial cone decompositions
zbMATH Open document ID: 1190.13026
author: Daniel Robertz
published in: Journal of Symbolic Computation
publication date: 2009-09-14T00:00:00Z
review text: Noether normalization is a cornerstone in computational commutative algebra. In the last two decades, many algorithms based on Gröbner bases and random changes of variables have been introduced by several authors. In this paper, the author uses Janet bases and the monomial cone decompositions they define to produce an effective implementation of a general algorithm for Noether normalization outlined by Vasconcelos in 1998 [\textit{W. V. Vasconcelos}, Computational methods of commutative algebra and algebraic geometry. Algorithms and Computation in Mathematics 2. Berlin: Springer (1988; Zbl 0896.13021)]. The main advantages in this novel use of Janet bases for Noether normalization are the detection of \textit{sparse} coordinate transformations and the ability to decide whether an ideal contains a monic polynomial in a given variable without computing elimination ideals (which is usually done via Gröbner bases). A suitable choice of coordinate transformation is crucial in this procedure to avoid computational bottlenecks arising from the fact that arbitrary transformations significantly increase the size of coefficients and the number of monomials in a given polynomial. Hence the procedure outlined in this paper has the promise of being significantly faster than most previous approaches. This is confirmed by an experiment performed by the author comparing his implementation against standard procedures found in some of the main computer algebra systems.
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 13F55
Mathematics Subject Classification ID: 13Pxx
zbMATH DE Number: 5603610
zbMATH Keywords: Noether normalization
zbMATH Keywords: involutive bases
zbMATH Keywords: Janet bases
zbMATH Keywords: Stanley decompositions
zbMATH Keywords: sparse coordinate change
reviewed by: Luis David Garcia Puente
describes a project that uses: noether.lib
describes a project that uses: SINGULAR
describes a project that uses: Macaulay2
describes a project that uses: Magma
describes a project that uses: Janet
MaRDI profile type: Publication
full work available at URL: j.jsc.2009.02.004
OpenAlex ID: W1984152394
cites work: The theory of involutive divisions and an application to Hilbert function computations
cites work: Saturation and Castelnuovo-Mumford regularity
cites work: The Magma algebra system. I: The user language
cites work: Bruno Buchberger's PhD thesis 1965: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal. Translation from the German
cites work: Characteristic-free bounds for the Castelnuovo–Mumford regularity
cites work: Q4247789
cites work: Q4317713
cites work: Finding sparse systems of parameters
cites work: Q5486584
cites work: Involutive bases of polynomial ideals
cites work: SINGULAR
cites work: Efficient Algorithms for Computing Nœther Normalization
cites work: An efficient algebraic algorithm for the geometric completion to involution
cites work: Q3826659
cites work: Q5735274
cites work: Janet's approach to presentations and resolutions for polynomials and linear PDEs
cites work: Q4303987
cites work: Q5431985
cites work: Q5307639
cites work: Combinatorics and commutative algebra.
cites work: Gröbner bases and Stanley decompositions of determinantal ideals
cites work: Computing combinatorial decompositions of rings
cites work: Computational methods of commutative algebra and algebraic geometry. With chapters by David Eisenbud, Daniel R. Grayson, Jürgen Herzog and Michael Stillman
DOI: 10.1016/J.JSC.2009.02.004
Recommended article: Q3826659
Recommended article: Q2760212
Recommended article: Q4815324
Recommended article: Efficient Algorithms for Computing Nœther Normalization
Recommended article: Q2760211
Recommended article: An algorithm for computing the integral closure
Recommended article: Q5486584
Recommended article: Generalized normal forms and polynomial system solving
Recommended article: Computing Gröbner Bases within Linear Algebra
Recommended article: Q4227325
description: scientific article; zbMATH DE number 5603610