On 3-matrix factorization of polynomials

Abstract

Let \(R=K[x_{1},x_{2},\cdots, x_{r}]\) and \(S=K[y_{1},y_{2},\cdots, y_{s}],\) where \(K\) is a field. 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 explicitly 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 \(m\times m\) matrices such that \(PQT=fI_{m}\). Moreover, we construct a bifunctorial operation \(\overline{\otimes}_{3}\) which is such that if \(X\) (respectively \(Y\)) is a \(3\)-matrix factorization of \(f\in R\) (respectively \(g\in S\)), then \(X\overline{\otimes}_{3} Y\) is a \(3\)-matrix factorization of \(fg\in K[x_{1},x_{2},\cdots, x_{r},y_{1},y_{2},\cdots, y_{s}]\). We call \(\overline{\otimes}_{3}\) the multiplicative tensor product of \(3\)-matrix factorizations. Finally, we give some properties of the operation \(\overline{\otimes}_{3}\).