On tensor products of matrix factorizations

TL;DR

This paper introduces a new multiplicative tensor product for matrix factorizations, enabling more efficient polynomial factorization algorithms with smaller matrix sizes.

Contribution

It proposes a novel bifunctorial tensor product for matrix factorizations that produces factorizations of the product of polynomials, improving existing algorithms.

Findings

Defined the multiplicative tensor product $ ilde{oxtimes}$ for matrix factorizations.

Proved properties of the new tensor product and its variants.

Developed an improved polynomial factorization algorithm with smaller matrix sizes.

Abstract

Let $K$ be a field. Let $f\in K[[x_{1},...,x_{r}]]$ and $g\in K[[y_{1},...,y_{s}]]$ be nonzero elements. If $X$ (resp. $Y$) is a matrix factorization of $f$ (resp. $g$), Yoshino had constructed a tensor product (of matrix factorizations) $\hat{\otimes}$ such that $X\hat{\otimes}Y$ is a matrix factorization of $f+g\in K[[x_{1},...,x_{r},y_{1},...,y_{s}]]$. In this paper, we propose a bifunctorial operation $\widetilde{\otimes}$ and its variant $\widetilde{\otimes}'$ such that $X\widetilde{\otimes}Y$ and $X\widetilde{\otimes}' Y$ are two different matrix factorizations of $fg\in K[[x_{1},...,x_{r},y_{1},...,y_{s}]]$. We call $\widetilde{\otimes}$ the multiplicative tensor product of $X$ and $Y$. Several properties of $\widetilde{\otimes}$ are proved. Moreover, we find three functorial variants of Yoshino's tensor product $\hat{\otimes}$. Then, $\widetilde{\otimes}$ (or its variant) is used in conjunction with $\hat{\otimes}$ (or any of its variants) to give an improved version of the standard algorithm for factoring polynomials using matrices on the class of summand-reducible polynomials defined in this paper. Our algorithm produces matrix factors whose size is at most one half the size one obtains using the standard method.