# Factorizations for a Class of Multivariate Polynomial Matrices

## TL;DR

This paper generalizes existing results on factorizing multivariate polynomial matrices, providing a new theorem and algorithm that expand the class of matrices that can be factorized, supported by an illustrative example.

## Contribution

The paper introduces a new theorem for factorizing polynomial matrices based on minors generating the unit ideal, extending previous work and offering a practical algorithm.

## Key findings

- The main theorem generalizes previous factorization conditions.
- An algorithm for factorizing a broader class of polynomial matrices is proposed.
- An example demonstrates the theorem's non-triviality and practical value.

## Abstract

Following the works by Lin et al. (Circuits Syst. Signal Process. 20(6): 601-618, 2001) and Liu et al. (Circuits Syst. Signal Process. 30(3): 553-566, 2011), we investigate how to factorize a class of multivariate polynomial matrices. The main theorem in this paper shows that an $l\times m$ polynomial matrix admits a factorization with respect to a polynomial if the polynomial and all the $(l-1)\times (l-1)$ reduced minors of the matrix generate the unit ideal. This result is a further generalization of previous works, and based on this, we give an algorithm which can be used to factorize more polonomial matrices. In addition, an illustrate example is given to show that our main theorem is non-trivial and valuable.