Piecewise Certificates of Positivity for matrix polynomials
Ronan Quarez (IRMAR)

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 admits a piecewise semi-certificate, i.e. a collection of identites where is a matrix polynomial and is a non negative polynomial on a semi-algebraic subset , where . 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 as the determinant of a positive semi-definite quadratic matrix polynomial.
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Taxonomy
TopicsMatrix Theory and Algorithms · Advanced Optimization Algorithms Research · Polynomial and algebraic computation
