On Factor Left Prime Factorization Problems for Multivariate Polynomial Matrices

TL;DR

This paper develops a comprehensive condition and algorithm for factor left prime factorizations of multivariate polynomial matrices without full row rank, with implementations demonstrating effectiveness through examples.

Contribution

It introduces a necessary and sufficient condition for factor left prime factorizations and provides an algorithm for computing all such factorizations.

Findings

Algorithm successfully computes factorizations in Maple

Conditions are both necessary and sufficient

Applicable to matrices without full row rank

Abstract

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.