On 3-matrix factorizations of polynomials

Abstract

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