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.
Categories: math.CT