This paper investigates how to factorize a class of multivariate polynomial matrices. We prove that an $$l\times m$$ l × m multivariate polynomial matrix admits a matrix factorization with respect to a given polynomial if the polynomial and all the $$(l-1)\times (l-1)$$ ( l - 1 ) × ( l - 1 ) reduced minors of the matrix generate a unit ideal. This result is a generalization of a theorem in Liu et al. (Circuits Syst Signal Process 30(3):553–566, 2011). Based on three main theorems presented in the paper and a constructive algorithm proposed by Lin et al. (Circuits Syst Signal Process 20(6):601–618, 2001), we give an algorithm which can be used to factorize more multivariate polynomial matrices. In addition, an illustrative example is given to show the effectiveness of the proposed algorithm.

中文翻译：

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一类多元多项式矩阵的因式分解

本文研究如何分解一类多元多项式矩阵。我们证明 $$l\times m$$ l × m 多元多项式矩阵允许对给定多项式进行矩阵分解，如果多项式和所有 $$(l-1)\times (l-1)$$ ( l - 1 ) × ( l - 1 ) 减少矩阵的次要生成单元理想。这个结果是 Liu 等人的定理的推广。（电路系统信号处理 30(3):553–566, 2011）。基于论文中提出的三个主要定理和 Lin 等人提出的构造算法。（Circuits Syst Signal Process 20(6):601–618, 2001），我们给出了一种算法，可用于分解更多的多元多项式矩阵。此外，给出了一个说明性的例子来说明所提出算法的有效性。