Pure states, positive matrix polynomials and sums of hermitian squares
نویسندگان
چکیده
منابع مشابه
Pure States, Positive Matrix Polynomials and Sums of Hermitian Squares
Let M be an archimedean quadratic module of real t× t matrix polynomials in n variables, and let S ⊆ R be the set of all points where each element of M is positive semidefinite. Our key finding is a natural bijection between the set of pure states of M and S × P(R). This leads us to conceptual proofs of positivity certificates for matrix polynomials, including the recent seminal result of Hol a...
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In recent years, much work has been devoted to a systematic study of polynomial identities certifying strict or non-strict positivity of a polynomial f on a basic closed set K ⊂ R. The interest in such identities originates not least from their importance in polynomial optimization. The majority of the important results requires the archimedean condition, which implies that K has to be compact....
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XS f dμ. It is natural to ask if the same is true for any linear functional L : R[X ] → R which is non-negative on MS. This is the Moment Problem for the quadratic module MS. The most interesting case seems to be when S is finite. A sufficient condition for it to be true is that each f ∈ T̃S can be approximated by elements of MS in the sense that there exists an element q ∈ R[X ] such that, for ...
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ژورنال
عنوان ژورنال: Indiana University Mathematics Journal
سال: 2010
ISSN: 0022-2518
DOI: 10.1512/iumj.2010.59.4107