To solving multiparameter problems of algebra. 9. The $\Psi F$-$q$ method for factorizing invariant polynomials and its applications

A new method (the $\Psi F$-$q$ method) for computing the invariant polynomials of a $q$-parameter $(q\ge1$) polynomial matrix $F$ is suggested. Invariant polynomials are computed in factored form, which permits one to analyze the structure of the regular spectrum of the matrix $F$, to isolate the divisors of each of the invariant polynomials whose zeros belong to the invariant polynomial in question, to find the divisors whose zeros belong to at least two of the neighboring invariant polynomials, and to determine the heredity levels of points of the spectrum for each of the invariant polynomials. Applications of the $\Psi F$-$q$ method to representing a polynomial matrix $F(\lambda)$ as a product of matrices whose spectra coincide with the zeros of the corresponding divisors of the characteristic polynomial and, in particular, with the zeros of an arbitrary invariant polynomial or its divisors are considered.