To solving multiparameter problems of algebra. 5. The $\nabla V$-$q$ factorization algorithm and its applications

The algorithm of $\nabla V$-factorization suggested earlier for decomposing one- and two-parameter polynomial matrices of full row rank into a product of two matrices (a regular one, whose spectrum coincides with the finite regular spectrum of the original matrix, and a matrix of full row rank, whose singular spectrum coincides with the singular spectrum of the original matrix, whereas the regular spectrum is empty) is extended to the case of $q$-parameter ($q\geqslant1$) polynomial matrices. The algorithm of $\nabla V$-$q$ factorization is described, and its justification and properties for matrices with arbitrary number of parameters are presented. Applications of the algorithm to computing irreducible factorizations of $q$-parameter matrices, to determining a free basis of the null-space of polynomial solutions of the matrix, and to finding matrix divisors corresponding to divisors of its characteristic polynomial are considered.