Real Algebraic and Analytic Geometry

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288. Igor Klep, Markus Schweighofer:
Pure states, positive matrix polynomials and sums of hermitian squares.

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Submission: 2009, September 29.

Let M be an archimedean quadratic module of real t-by-t matrix polynomials in n variables, and let S be the set of all n-tuples where each element of M is positive semidefinite. Our key finding is a natural bijection between the set of pure states of M and the cartesian product of S with the real projective (t-1)-space. This leads us to conceptual proofs of positivity certificates for matrix polynomials, including the recent seminal result of Hol and Scherer: If a symmetric matrix polynomial is positive definite on S, then it belongs to M. We also discuss what happens for non-symmetric matrix polynomials or in the absence of the archimedean assumption, and review some of the related classical results. The methods employed are both algebraic and functional analytic.

Mathematics Subject Classification (2000): 15A48, 11E25, 13J30, 15A54, 14P10, 46A55.

Keywords and Phrases: matrix polynomial, pure state, positive semidefinite matrix, sum of hermitian squares, Positivstellensatz, archimedean quadratic module, Choquet theory.

Full text, 15p.: dvi 139k, ps.gz 348k, pdf 367k.

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