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Abstract. 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) = P j fi,j(x)Ui,j(x) T Ui,j(x) where Ui,j(x) is a matrix polynomial and fi,j(x) is a non negative polynomial on a semialgebraic subset Si, where Rn = ∪r i=1Si. 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. 1

Year: 2010

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