This paper is concerned with factor left prime factorization problems for multivariate polynomial matrices without full row rank. We propose a necessary and sufficient condition for the existence of factor left prime factorizations of a class of multivariate polynomial matrices, and then design an algorithm to compute all factor left prime factorizations if they exist. We implement the algorithm on the computer algebra system Maple, and two examples are given to illustrate the effectiveness of the algorithm. The results presented in this paper are also true for the existence of factor right prime factorizations of multivariate polynomial matrices without full column rank.

本文涉及没有全行秩的多元多项式矩阵的因子左素数分解问题。我们为一类多元多项式矩阵的因子左素因式分解的存在提出了充要条件，然后设计一种算法来计算所有因子左素因式分解（如果存在）。我们在计算机代数系统Maple上实现了该算法，并给出了两个例子来说明该算法的有效性。本文提出的结果对于没有完整列等级的多元多项式矩阵的因数素数分解也同样适用。