Piecewise Certificates of Positivity for matrix polynomials

Abstract

We show that any symmetric positive definite homogeneous matrix polynomial $M\in \mathbb{R}[x_1,...,x_n{]}^{m\times m}$ admits a piecewise semi-certificate, i.e. a collection of identites $M(x)={\sum }_j{f}_{i,j}(x)U_{i,j}(x{)}^T{U}_{i,j}(x)$ where $U_{i,j}(x)$ is a matrix polynomial and $f_{i,j}(x)$ is a non negative polynomial on a semi-algebraic subset $S_i$, where ${\mathbb{R}}^n={\cup }_{i=1}^r{S}_i$. This result generalizes to the setting of biforms. Some examples of certificates are given and among others, we study a variation around the Choi counterexample of a positive semi-definite biquadratic form which is not a sum of squares. As a byproduct we give a representation of the famous non negative sum of squares polynomial $x^4{z}^2+z^4{y}^2+y^4{x}^2-3x^2{y}^2{z}^2$ as the determinant of a positive semi-definite quadratic matrix polynomial.