Multivariate Polynomials, Duality, and Structured Matrices

We rst review the basic properties of the well known classes of Toeplitz Hankel Vandermonde and other related structured matrices and re examine 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 compu tations and to improve substantially the known techniques of the study of these problems In particular we simplify and or generalize the known reduction of the multivariate polynomial systems to matrix eigenproblem the derivation of the B zout and Bernshtein bounds on the number of the roots and the construction of multiplication tables From the algorithmic and computational complexity point we yield acceleration by one order of magnitude of the known methods for some fundamental problems of solving multivariate polynomial systems of equations