Multivariate Polynomials, Duality, and Structured Matrices

We rst review thebasic properties of the well knownclasses of Toeplitz, Hankel, Vandermonde, and other related structured matrices and reexamine their correlation to operations with univariate polynomials. Then we de ne some natural extensions of such classes of matrices based on their correlation to multivariate polynomials. We describe the correlation in terms of the associated operators of multiplication in the polynomial ring and its dual space, which allows us to generalize these structures to the multivariate case. Multivariate Toeplitz, Hankel, and Vandermonde matrices, Bezoutians, algebraic residues and relations between them are studied. Finally, we show some applications of this study to root nding problems for a system of multivariate polynomial equations, where the dual space, algebraic residues, Bezoutians and other structured matrices play an important role. The developed techniques enable us to obtain a better insight into the major problems of multivariate polynomial c...

• Type: Journal
• Year: 2000
• Where: JC
• Authors: Bernard Mourrain, Victor Y. Pan
• Keywords: JC 2000, Multivariate, Multivariate Polynomial, Polynomial