Piecewise Certificates of Positivity for matrix polynomials

Ronan Quarez (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

We show that any symmetric positive definite homogeneous matrix polynomial $M\in\R[x_1,\ldots,x_n]^{m\times m}$ 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.\par 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.

Mots clés: Biforms; Matrix polynomials; Positive semi-definite; Positivity certificate; Sum of Squares

Domaines: Autre [cs.OH]; Anneaux et algèbres [math.RA]

Ronan Quarez. Piecewise Certificates of Positivity for matrix polynomials. 2010. ⟨hal-00445256⟩