Using symmetries in the eigenvalue method for polynomial systems
type: Item
instance of: scholarly article
title: Using symmetries in the eigenvalue method for polynomial systems
zbMATH Open document ID: 1174.14053
author: Karin Gatermann
author: Ilias S. Kotsireas
author: Robert M. Corless
published in: Journal of Symbolic Computation
publication date: 2009-10-09T00:00:00Z
review text: Let \(I\subset \mathbb{C}[x_1, \dots, x_n]\) be a zero--dimensional ideal. For \(f\in\mathbb{C}[x_1, \dots, x_n]\) denote by \(m_f\) the matrix defined by the multiplication with \(f\) in the algebra \(\mathbb{C}[x_1, \dots, x_n]/I\) and \(h_f\) its minimal polynomial. It is known that the roots of \(h_f=0\) coincide with the eigenvalues of \(m_f\) resp. the values of \(f\) on \(V(I)\). Using this fact one can compute the eigenvalues numerically and find the points in \(V(I)\), i.e. the solutions of the polynomial equations defining \(I\). This so-called eigenvalue method is studied and adapted to the case that the system of polynomial equations has symmetries. It is proved that the corresponding matrices occuring in this situation have a special block diagonal structure. An algorithm is given to exploit this result in an efficient way.
reviewed by: Gerhard Pfister
Mathematics Subject Classification ID: 14Q99
Mathematics Subject Classification ID: 13P10
zbMATH DE Number: 5611764
zbMATH Keywords: solving polynomial systems
zbMATH Keywords: eigenvalue method
zbMATH Keywords: symmetries
describes a project that uses: SINGULAR
describes a project that uses: Macaulay2
describes a project that uses: Maple
MaRDI profile type: Publication
full work available at URL: j.jsc.2008.11.009
OpenAlex ID: W1968321788
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DOI: 10.1016/J.JSC.2008.11.009
description: scientific article; zbMATH DE number 5611764