To solving multiparameter problems of algebra. 7. The $PG$-$q$ factorization method and its applications

The paper continues the development of rank-factorization methods for solving certain algebraic problems for multiparameter polynomial matrices and introduces a new rank factorization of a $q$-parameter polynomial $m\times n$ matrix $F$ of full row rank (called the $PG$-$q$ factorization) of the form $F=PG$, where $P=\prod\limits^{q-1}_{k=1}\prod\limits^{n_k}_{i=1}\nabla^{(k)}_i$ is the greatest left divisor of $F$; $\nabla^{(k)}_i$ is a regular $(q-k)$-parameter polynomial matrix the characteristic polynomial of which is a primitive polynomial over the ring of polynomials in $q-k-1$ variables, and $G$ is a $q$-parameter polynomial matrix of rank $m$. The $PG$-$q$ algorithm is suggested, and its applications to solving some problems of algebra are presented. Bibliography: 6 titles.