Some properties of n-matrix factorizations of polynomials

TL;DR

This paper investigates properties of n-matrix factorizations of polynomials over polynomial rings, providing bounds on the number of factors and analyzing specific cases like sums of squares.

Contribution

It introduces new properties of n-matrix factorizations and establishes lower bounds on the number of factors needed, especially for sums of squares polynomials.

Findings

Derived lower bounds on the number of n-matrix factors

Analyzed properties of n-matrix factorizations in polynomial rings

Provided specific bounds for sums of squares polynomial with m=8

Abstract

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 $m=8$.