A New Algorithm for General Factorizations of Multivariate Polynomial Matrices

We investigate how to factorize a multivariate polynomial matrix into the product of two matrices. There are two major parts. The first is a factorization theorem, which asserts that a multivariate polynomial matrix whose lower order minors satisfy certain conditions admits a matrix factorization. Our theory is a generalization to the previous results given by Lin et.al [16] and Liu et.al [17]. The second is the implementation for factorizing polynomial matrices. According to the proof of factorization theorem, we construct a main algorithm which extends the range of polynomial matrices that can be factorized. In this algorithm, two critical steps are involved in how to compute a zero left prime matrix and a unimodular matrix. Firstly, based on the famous Quillen-Suslin theorem, a new sub-algorithm is presented to obtain a zero left prime matrix by calculating the bases of the syzygies of two low-order polynomial matrices. Experiments show that it is more efficient than the algorithm constructed by Wang and Kwong [31]. Secondly, some auxiliary information provided by the above new sub-algorithm is used to construct a unimodular matrix. As a consequence, the main algorithm extends the application range of the constructive algorithm in [17]. We implement all the algorithms proposed above on the computer algebra system Singular and give a nontrivial example to show the process of the main algorithm.