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arXiv : 1001.1277

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Piecewise Certificates of Positivity for matrix polynomials

Ronan Quarez 1

(07/01/2010)

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.

1 :	Institut de Recherche Mathématique de Rennes (IRMAR)
	CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne

Domaine

:

Informatique/Autre Mathématiques/Anneaux et algèbres

Mots Clés : Biforms – Matrix polynomials – Positive semi-definite – Positivity certificate – Sum of Squares

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Contributeur : Ronan Quarez <>

Soumis le : Vendredi 8 Janvier 2010, 09:13:49

Dernière modification le : Mercredi 24 Mars 2010, 10:57:27