2024/02/01 by Yves Fomatati, Fomatati, Yves Baudelaire
Computer Science · Mathematics · #15A23 #18A05 #Advanced Optimization Algorithms Research #Algebraic and Geometric Analysis #Category Theory (math.CT) #FOS: Mathematics #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.2402.00991
openalex publication_date 2024/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let R=K[x1,x2,⋯, xm] and S= K[y1,y2,⋯, ym] where K is a field. %commutative ring with unity. In this paper, we propose a method showing how to obtain 3-matrix factors for a given polynomial using either the Doolittle or the Crout decomposition techniques that we apply to matrices whose entries are not real numbers but polynomials. We also define the category of 3-matrix factorizations of a polynomial f whose objects are 3-matrix factorizations of f, that is triplets (P,Q,T) of m× m matrices such that PQT=fIm. Moreover, we construct a bifunctorial operation ⊗3 which is such that if X (respectively Y) is a 3-matrix factorization of f∈ R (respectively g∈ S), then X⊗3 Y is a 3-matrix factorization of fg∈ K[x1,x2,⋯, xm,y1,y2,⋯, ym]. We call ⊗3 the multiplicative tensor product of 3-matrix factorizations. Finally, we give some properties of the operation ⊗3.