The multiparameter (including one-parameter) polynomial and rational matrices (as regular, as singular) and vector spaces are the investigation object. The definitions of the base spectral characteristics (a polynomial basis of a null-space of a matrix; the finite and "infinite" spectrum, the regular and singular parts of the spectrum, an eigenvector and a Jordan semilattice of the vectors of a polynomial matrix; a pole-zero structure of a rational matrix) for the multiparameter matrices, consistent with the classical definitions for one-parameter case, were introduced. The generalized definitions: an eigenpolynomial, the generating eigenvector and principal vector, a free basis were introduced. The properties of these characteristics were investigated. The linearization of a polynomial matrix, realized by passage to an accompanying pencil of the constant matrices, is proposed. The representations of a rational matrix in the form MFD and PFD are considered, their properties are investigated. The rank factorization methods for the polynomial and rational matrices, which allow solving some partial multiparameter spectral problems, are developed. The methods for solving the multiparameter problems of algebra based on the resultant approach are developed. The methods of solving "connected" multiparameter spectral problems, are developed. The applications of the mentioned problems are particularly the problem of solving systems of nonlinear algebraic and rational equations and the inverse eigenvalue problems for a matrix (including additive and multiplicative problems).
Main publications:
V. B. Khazanov, “METHODS FOR SOLVING SOME PARAMETRIC PROBLEMS OF ALGEBRA”, JOURNAL OF MATHEMATICAL SCIENCES, 137:3 (2006), 4852-4861
V.B. Khazanov, “THE RESULTANT APPROACH TO COMPUTING VECTOR CHARACTERISTICS OF MULTIPARAMETER POLYNOMIAL MATRICES”, JOURNAL OF MATHEMATICAL SCIENCES, 137:3 (2006), 4862-4878
V. B. Khazanov, “ON SOME SPECTRAL CHARACTERISTICS OF MULTIPARAMETER POLYNOMIAL MATRICES”, JOURNAL OF MATHEMATICAL SCIENCES, 132:2 (2006), 236-239
V.B. Khazanov, “ON SOME PROPERTIES OF POLYNOMIAL BASES OF SUBSPACES OVER THE FIELD OF RATIONAL FUNCTIONS IN SEVERAL VARIABLES”, JOURNAL OF MATHEMATICAL SCIENCES, 121:4 (2004), 2538-2545
V.B. Khazanov, “BALANCE RELATION FOR THE SPECTRAL CHARACTERISTICS OF A MULTIPARAMETER POLYNOMIAL MATRIX”, JOURNAL OF MATHEMATICAL SCIENCES, 165:5 (2010), 597-600
