A Property of Modules Over a Polynomial Ring With an Application in Multivariate Polynomial Matrix Factorizations

This paper is concerned with a property of modules over a polynomial ring and its application in multivariate polynomial matrix factorizations. We construct a specific polynomial such that the product of the polynomial and a nonzero vector in a module over a polynomial ring can be represented by the elements in a maximum linearly independent vector set of the module over the polynomial ring. Based on this property, a relationship between a rank-deficient matrix and any of its full row rank submatrices is presented. By this result, we show that the problem for general factorizations of rank-deficient matrices can be translated into that of any of their full row rank submatrices in the regular case. Then many results on factorizations of full row rank matrices, such as zero prime factorizations, minor prime factorizations and factor prime factorizations, can be extended to the rank-deficient case. We implement the algorithm of general factorizations for rank-deficient matrices on the computer algebra system Maple, and two examples are given to illustrate the algorithm.