Multivariate polynomial equations with multiple zeros solved by matrix eigenproblems
type: Item
instance of: scholarly article
title: Multivariate polynomial equations with multiple zeros solved by matrix eigenproblems
zbMATH Open document ID: 0851.65029
author: Hans J. Stetter
author: Hans-Michael Möller
published in: Numerische Mathematik
publication date: 1996-11-26T00:00:00Z
review text: The paper deals with the eigenproblem approach for solving systems of polynomial equations. In particular, the case of isolated zeros only is considered. From a Gröbner basis of the zero-dimensional ideal generated by the polynomial equations an \(n\times n\)-matrix \(A_f\), depending on some suitable polynomial \(f\), can be computed. Here \(n\) denotes the number of zeros counted with multiplicity. The case of simple zeros only, which is shortly reviewed, leads to the classical eigenvalue/eigenvector problem for \(A_f\). The intention of the present paper is to extend the theory to the case of multiple zeros. It results in a generalization of the Jordan normal form decomposition of \(A_f\). The case of nonderogatory matrices \(A_f\) is specially addressed. An algorithmic analysis of the eigenproblem in the general case is indicated.
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 12Y05
Mathematics Subject Classification ID: 26C10
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 12D10
Mathematics Subject Classification ID: 30C15
Mathematics Subject Classification ID: 65H17
zbMATH DE Number: 770162
zbMATH Keywords: matrix eigenproblems
zbMATH Keywords: eigenvalue
zbMATH Keywords: eigenvector
zbMATH Keywords: eigenproblem approach
zbMATH Keywords: systems of polynomial equations
zbMATH Keywords: isolated zeros
zbMATH Keywords: Gröbner basis
zbMATH Keywords: multiple zeros
zbMATH Keywords: Jordan normal form decomposition
zbMATH Keywords: nonderogatory matrices
reviewed by: Walter Zulehner
MaRDI profile type: Publication
full work available at URL: s002110050122
OpenAlex ID: W2030143626
DOI: 10.1007/S002110050122
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description: scientific article; zbMATH DE number 770162