Generalized characteristic polynomials

KYBERNETIKA-PRAHA-, 2003

Journal of Complexity, 2000

We rst review the basic properties of the well known classes 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 computations 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.

Publications of the Research Institute for Mathematical Sciences, 1985

This paper derives a determinant form formula for the general solution of coupled linear equations with coefficients in K[XI, ....... x n ], where K is a field of numbers, the number of unknowns is greater than the number of equations, and the solutions are in K(x\, ..., Xn-i)[x n ]-The formula represents the general solution by the minimum number of generators, and it is a generalization of Cramer's formula for the solutions in K(XI, ..., x n)-Compared with another formula which is obtained by a method typical in algebra, the generators in our formula are represented by determinants of quite small orders.

Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation

In this paper we obtain new results on multivariate dimension polynomials of differential field extensions associated with partitions of basic sets of derivations. We prove that the coefficient of the summand of the highest possible degree in the canonical representation of such a polynomial is equal to the differential transcendence degree of the extension. We also give necessary and sufficient conditions under which the multivariate dimension polynomial of a differential field extension of a given differential transcendence degree has the simplest possible form. Furthermore, we describe some relationships between a multivariate dimension polynomial of a differential field extension and dimensional characteristics of subextensions defined by subsets of the basic sets of derivations. CCS CONCEPTS • Computing methodologies → Symbolic and algebraic manipulation.

Chinese Annals of Mathematics, Series B, 2007

A linear system arising from a polynomial problem in the approximation theory is studied, and the necessary and sufficient conditions for existence and uniqueness of its solutions are presented. Together with a class of determinant identities, the resulting theory is used to determine the unique solution to the polynomial problem. Some homogeneous polynomial identities as well as results on the structure of related polynomial ideals are just by-products.