Some properties of n-matrix factorizations of polynomials

Let $R=K[x_1,x_2,\cdots ,x_m]$ where $K$ is a field. In this paper, we give some properties of $n$-matrix factorizations of polynomials in $R$. We also derive some results giving some lower bounds on the number of $n$-matrix factors of polynomials. In particular, we give a lower bound on the number of matrix factors of minimal size for the sums of squares polynomial $f_m=x_1^2+\cdots +x_m^2$ for $m=8$.