On tensor products of matrix factorizations

Abstract

Let K be a field. Let f∈K[[x1,...,xr]] and g∈K[[y1,...,ys]] be nonzero elements. If X (resp. Y) is a matrix factorization of f (resp. g), Yoshino had constructed a tensor product (of matrix factorizations) ⊗ˆ such that X⊗ˆY is a matrix factorization of f+g∈K[[x1,...,xr,y1,...,ys]]. In this paper, we propose a bifunctorial operation ⊗˜ and its variant ⊗˜′ such that X⊗˜Y and X⊗˜′Y are two different matrix factorizations of fg∈K[[x1,...,xr,y1,...,ys]]. We call ⊗˜the multiplicative tensor product of X and Y. Several properties of ⊗˜ are proved. Moreover, we find three functorial variants of Yoshino's tensor product ⊗ˆ. Then, ⊗˜ (or its variant) is used in conjunction with ⊗ˆ (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.