Refined multiplicative tensor product of matrix factorizations

TL;DR

This paper introduces a refined multiplicative tensor product for matrix factorizations that improves an existing polynomial factorization algorithm, resulting in smaller matrix factors for summand-reducible polynomials.

Contribution

The paper proposes a new reduced multiplicative tensor product of matrix factorizations, enhancing an existing algorithm for polynomial matrix factorization by producing smaller matrix factors.

Findings

Refined tensor product yields smaller matrix factors.

Improved algorithm performs better on summand-reducible polynomials.

Enhanced method outperforms standard approaches.

Abstract

An algorithm for matrix factorization of polynomials was proposed in \cite{fomatati2022tensor} and it was shown that this algorithm produces better results than the standard method for factoring polynomials on the class of summand-reducible polynomials. In this paper, we improve this algorithm by refining the construction of one of its two main ingredients, namely the multiplicative tensor product $\widetilde{\otimes}$ of matrix factorizations to obtain another different bifunctorial operation that we call the reduced multiplicative tensor product of matrix factorizations denoted by $\overline{\otimes}$. In fact, we observe that in the algorithm for matrix factorization of polynomials developed in \cite{fomatati2022tensor}, if we replace $\widetilde{\otimes}$ by $\overline{\otimes}$, we obtain better results on the class of summand-reducible polynomials in the sense that the refined algorithm produces matrix factors which are of smaller sizes.