Construction of Gröbner bases for investigation of systems of polynomial equations
type: Item
instance of: scholarly article
title: Construction of Gröbner bases for investigation of systems of polynomial equations
zbMATH Open document ID: 0927.13025
DOI: 10.1007/BF02667196
author: A. Yu. Rystsova
author: I. K. Rystsov
published in: Cybernetics and Systems Analysis
publication date: 1999-04-28T00:00:00Z
review text: In the first part of this article, the authors give a quick introduction to Gröbner bases, limiting themselves to the definitions and facts necessary to understand how Gröbner bases can be employed to help find the roots of systems of multivariate polynomials over the rationals. The remainder of the article consists of an investigation of two small systems, the Katsura 3 and a set of 4 equations (3 quadratic and 1 linear) in 4 unknowns. With the help of the examples the authors make several points about how the ordering on the variables can affect the information obtainable from the resulting Gröbner basis. While some orderings may yield polynomials which can be completely solved using numerical methods, others may give polynomials which cannot be solved by known methods, polynomials such that the effects of numerical inaccuracies are greatly compounded as the values of successive variables are found, or polynomials yielding very little information about even the number of roots of the system. Very little is said about how to choose the best ordering on the variables.
reviewed by: Alyson Reeves
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 68W30
zbMATH DE Number: 1262961
zbMATH Keywords: nonlinear equations
zbMATH Keywords: Gröbner bases
zbMATH Keywords: numerical methods
zbMATH Keywords: numerical analysis
zbMATH Keywords: systems of multivariate polynomials
describes a project that uses: Maple
MaRDI profile type: Publication
cites work: Q3750117
cites work: Q5648448
cites work: Some examples for solving systems of algebraic equations by calculating Gröbner bases
cites work: Q4144192
cites work: Q4002478
description: scientific article; zbMATH DE number 1262961