Solving zero-dimensional algebraic systems
type: Item
instance of: scholarly article
title: Solving zero-dimensional algebraic systems
zbMATH Open document ID: 0753.13012
DOI: 10.1016/S0747-7171(08)80086-7
author: D. Lazard
published in: Journal of Symbolic Computation
publication date: 1992-09-26T00:00:00Z
review text: Let \(X_ 1,\ldots,X_ n\) be \(n\) variables ordered as \(X_ 1<X_ 2<\cdots<X_ n\). The main variable of a polynomial is the greatest variable in it. A set of \(n\) polynomials is called \textit{triangular} if the main variable of the \(i\)-th polynomial is \(X_ i\) and monic in \(X_ i\). Using this terminology, the author first remarks: Proposition 1. If \(K\) is a field, any maximal ideal in \(K[X_ 1,\ldots,X_ n]\) has a triangular system of generators. Proposition 2. Every system with a finite number of solutions (in an algebraic closure of \(K\)) is equivalent to the union of a finite number of triangular systems. Then he shows how to compute numerically the common zeros of a triangular set.
Mathematics Subject Classification ID: 13P05
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 68W30
zbMATH DE Number: 56190
zbMATH Keywords: triangular set
zbMATH Keywords: compute numerically the common zeros
zbMATH Keywords: Gröbner base
Wikidata QID: Q55883678
reviewed by: Jun Nishimura
MaRDI profile type: Publication
DBLP publication ID: journals/jsc/Lazard92
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description: scientific article; zbMATH DE number 56190