On Minor Left Prime Factorization Problem for Multivariate Polynomial Matrices

TL;DR

This paper introduces a new condition for factorizing multivariate polynomial matrices without full row rank, along with an algorithm implemented in Maple that demonstrates efficiency through illustrative examples.

Contribution

It provides a novel necessary and sufficient condition for minor left prime factorizations and develops an efficient algorithm based on this condition.

Findings

The algorithm successfully factorizes matrices in tested examples.

Experimental data confirms the algorithm's efficiency.

The method extends factorization techniques to matrices without full row rank.

Abstract

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.