Multivariate polynomials, duality, and structured matrices
type: Item
instance of: scholarly article
title: Multivariate polynomials, duality, and structured matrices
zbMATH Open document ID: 0963.68232
published in: Journal of Complexity
publication date: 2001-06-12T00:00:00Z
full work available at URL: RR-3513.pdf
review text: The authors study solutions of systems of polynomial equations in several variables. They generalize Toeplitz, Hankel, Vandermonde, and other structured matrices to several variables and the relation to operations with polynomials. The authors also study Bezoutians, algebraic residues and relations between them. They apply the theory to rootfinding problems for systems of multivariate polynomial equations. The dual space, algebraic residues, Bezoutians and other structured matrices have an important role. Their algorithms improve over the previous methods, by an order of magnitude, the computational complexity of some problems. They give an overview of the univariate case followed by a generalization to multivariate polynomials. Let \(R=\mathbb C [x_1,x_2, \dots{}, x_n]\) be the ring of complex polynomials in the variables \(x_1, x_2, \dots{}, x_n\). Suppose \(A=R/I\) where \(I\) is the ideal generated by a finite set of polynomials with only finitely many roots \(\zeta_1, \zeta_2, \dots{}, \zeta_d\). \(R^*\) and \(A^*\) denote the dual spaces of linear functionals of \(R,A\) respectively. \(A^*\) can be identified with the the members of \(R^*\) that vanish on the ideal \(I\). The operator of multiplying by \(a\in A\) is denoted \(M_a\). Theorem 1: The set of eigenvalues of \(M_a\) is precisely \(\{ a(\zeta_1), a(\zeta_2), \dots{}, a(\zeta_d)\} \). Theorem 2: The common eigenvectors of the dual operator \({M_a}^*\) in \(A^*\) are, up to a scalar factor, the evaluations at \(\zeta_1, \zeta_2, \dots{}, \zeta_d\). Let \(E,F\) be finite subsets of \({\mathbb N}^n\). \(M\) is an \(E\times F\)-quasi-Toeplitz matrix if \(m_{\alpha,\beta }= m_{\gamma, \delta }\) provided \(\alpha - \beta = \gamma - \delta \). A matrix \(M\) is a quasi-Hankel if \(m_{\alpha, \beta } = m_{\gamma, \delta }\) whenever \(\alpha + \beta = \gamma + \delta \). Multiplication of a quasi-Toeplitz (-Hankel) matrix by a vector can be interpreted as (Laurent's) polynomial multiplication. The authors al…
reviewed by: Awad A. Iskander
Mathematics Subject Classification ID: 68W30
zbMATH DE Number: 1444123
author: Bernard Mourrain
author: Victor Y. Pan
MaRDI profile type: Publication
OpenAlex ID: W2011114755
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DOI: 10.1006/JCOM.1999.0530
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description: scientific article; zbMATH DE number 1444123