Real Algebraic and Analytic Geometry |

Pure states, positive matrix polynomials and sums of hermitian squares.

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Submission: 2009, September 29.

*Abstract:
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.

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