On 3-matrix factorizations of polynomials

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 $PQT=f{I}_m$. 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 ${\bar{\otimes }}_3$.