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Abstract : |
We first review the basic properties of the well known classes of Toeplitz, Hankel, Vandermonde, and other related structured matrices and re-examine their correlations to operations with univariate polynomials. Then we define some natural extensions of such classes of matrices based on their correlations to multivariate polynomials. We describe these correlations 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 among them are studied. Finally, we show some applications of structured matrices, duality and residues to rootfinding problems for a system of multivariate polynomial equations, where the dual space of linear forms, 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 computations and to improve substantially the known techniques of the study of these major problems. From the algorithmic point, we yield acceleration by order of magnitude of the known methods for some fundamental problems of solving multivariate polynomial systems of equations., |
