On tensor products of matrix factorizations

Abstract

Let K be a field. Let f ∈ K[[x 1 , ..., x r ]] and g ∈ 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) ⊗ such that X ⊗Y is a matrix factorization of f + g ∈ K[[x 1 , ..., x r , y 1 , ..., y s ]]. In this paper, we propose a bifunctorial operation ⊗ and its variant ⊗ ′ such that X ⊗Y and X ⊗ ′ Y are two different matrix factorizations of f g ∈ K[[x 1 , ..., x r , y 1 , ..., y s ]]. 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 ⊗. Then, ⊗ (or its variant) is used in conjunction with ⊗ (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.

Necessary conditions for the existence of Morita Contexts in the bicategory of Landau-Ginzburg Models

“…The standard algorithm to factor polynomials using matrices is found in [10] and for an improved version see [14] and [15].…”

“…Yoshino's tensor product has three mutually distinct functorial variants as can be seen in [15] 3 Properties of Matrix Factorizations…”

Some applications of the multiplicative tensor product of matrix factorizations

“…They improved the standard method for factoring polynomials for this class of polynomials. More recently in 2019, the author in his Ph.D. dissertation [8] defined the multiplicative tensor product of matrix factorizations and found a variant of this product [9] in 2020. These were then used to further improve the standard method for factoring a large class of polynomials.…”