To solving multiparameter problems of algebra. 4. The $AB$-algorithm and its applications

The paper continues investigations of methods for factorizing $q$-parameter polynomial matrices and considers their applications to solving multiparameter problems of algebra. An extension of the $AB$-algorithm, suggested earlier as a method for solving spectral problems for matrix pencils of the form $A-\lambda B$, to the case of $q$-parameter ($q\geqslant1$) polynomial matrices of full rank is proposed. In accordance with the $AB$-algorithm, a finite sequence of $q$-parameter polynomial matrices such that every subsequent matrix provides a basis of the null-space of polynomial solutions of its transposed predecessor is constructed. A certain rule for selecting specific basis matrices is described. Applications of the $AB$-algorithm to computing hereditary polynomials of a $q$-parameter polynomial matrix and exhausting them from the regular spectrum of the matrix, to constructing irreducible factorizations of rational matrices satisfying certain assumptions, and to computing “free” bases of the null-spaces of polynomial solutions of an arbitrary $q$-parameter polynomial matrix are considered.