Piecewise Certificates of Positivity for matrix polynomials

TL;DR

This paper introduces a piecewise semi-certificate approach for representing symmetric positive definite matrix polynomials as sums of squares on semi-algebraic sets, extending to biforms and providing new certificates and representations.

Contribution

It generalizes certificates of positivity for matrix polynomials to a piecewise setting and offers new examples and representations, including a notable non-negative polynomial as a determinant.

Findings

Any symmetric positive definite homogeneous matrix polynomial admits a piecewise semi-certificate.

The approach extends to biforms and includes examples related to Choi's counterexample.

A new representation of a famous non-negative polynomial as a determinant of a positive semi-definite matrix polynomial.

Abstract

We show that any symmetric positive definite homogeneous matrix polynomial $M\in\R[x_1,...,x_n]^{m\times m}$ admits a piecewise semi-certificate, i.e. a collection of identites $M(x)=\sum_jf_{i,j}(x)U_{i,j}(x)^TU_{i,j}(x)$ 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 $\R^n=\cup_{i=1}^r S_i$. 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 $x^4z^2+z^4y^2+y^4x^2-3 x^2y^2z^2$ as the determinant of a positive semi-definite quadratic matrix polynomial.