Thumbnail
Access Restriction
Open

Author Quarez, Ronan
Source arXiv.org
Content type Text
File Format PDF
Date of Submission 2010-01-08
Language English
Subject Domain (in DDC) Computer science, information & general works ♦ Natural sciences & mathematics ♦ Mathematics
Subject Keyword Mathematics - Rings and Algebras ♦ Computer Science - Other Computer Science ♦ cs ♦ math
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.
Educational Use Research
Learning Resource Type Article


Open content in new tab

   Open content in new tab