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Piecewise Certificates of Positivity for matrix polynomials
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
More on Rotations as Spin Matrix Polynomials
Any nonsingular function of spin j matrices always reduces to a matrix
polynomial of order 2j. The challenge is to find a convenient form for the
coefficients of the matrix polynomial. The theory of biorthogonal systems is a
useful framework to meet this challenge. Central factorial numbers play a key
role in the theoretical development. Explicit polynomial coefficients for
rotations expressed either as exponentials or as rational Cayley transforms are
considered here. Structural features of the results are discussed and compared,
and large j limits of the coefficients are examined.Comment: Additional references, simplified derivation of Cayley transform
polynomial coefficients, resolvent and exponential related by Laplace
transform. Other minor changes to conform to published version to appear in J
Math Phy
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