To solving problems of algebra for two-parameter polynomial matrices. 1

This paper starts a series of publications devoted to surveying and developing methods for solving algebraic problems for two-parameter polynomial and rational matrices. The paper considers rank factorizations, in particular, the relatively irreducible factorization and $\Delta W$-2 factorization, which are used in solving spectral problems for two-parameter polynomial matrices $F(\lambda,\mu)$. Algorithms for computing these factorizations are suggested and applied to computing points of the regular, singular, and regular-singular spectra and the corresponding spectral vectors of $F(\lambda,\mu)$. The computation of spectrum points reduces to solving algebraic equations in one variable. A new method for computing spectral vectors for given spectrum points is suggested. Algorithms for computing critical points and for constructing a relatively free basis of the right null-space of $F(\lambda,\mu)$ are presented. Conditions sufficient for the existence of a free basis are established, and algorithms for checking them are provided. An algorithm for computing the zero-dimensional solutions of a system of nonlinear algebraic equations in two variables is presented. The spectral properties of the $\Delta W$-2 method are studied. Bibl. – 4 titles.