2010/01/08 by Ronan Quarez, Quarez, Ronan
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #FOS: Computer and information sciences #FOS: Mathematics #Matrix Theory and Algorithms #Other Computer Science (cs.OH) #Polynomial and algebraic computation #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.1001.1277
openalex publication_date 2010/01/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that any symmetric positive definite homogeneous matrix polynomial M∈\R[x1,...,xn]m× m admits a piecewise semi-certificate, i.e. a collection of identites M(x)=∑jfi,j(x)Ui,j(x)TUi,j(x) where Ui,j(x) is a matrix polynomial and fi,j(x) is a non negative polynomial on a semi-algebraic subset Si, where \Rn=∪i=1r Si. 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 x4z2+z4y2+y4x2-3 x2y2z2 as the determinant of a positive semi-definite quadratic matrix polynomial.