A new necessary and sufficient condition for the existence of minor left prime factorizations of multivariate polynomial matrices without full row rank is presented. The key idea is to establish a relationship between a matrix and its full row rank submatrix. Based on the new result, we propose an algorithm for factorizing matrices and have implemented it on the computer algebra system Maple. Two examples are given to illustrate the effectiveness of the algorithm, and experimental data shows that the algorithm is efficient.

中文翻译：

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多元多项式矩阵的次左素因数分解问题

给出了无满行秩的多元多项式矩阵的次左素因分解存在的一个新的充要条件。关键思想是建立矩阵与其全行秩子矩阵之间的关系。基于新的结果，我们提出了一种矩阵分解算法，并在计算机代数系统 Maple 上实现了它。给出了两个例子来说明算法的有效性，实验数据表明该算法是有效的。