Real Algebraic and Analytic Geometry

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356. Vincent Astier, Thomas Unger:
Signatures of hermitian forms, positivity, and an answer to a question of Procesi and Schacher.

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Submission: 2016, September 27.

Using the theory of signatures of hermitian forms over algebras with involution, developed by us in earlier work, we introduce a notion of positivity for symmetric elements and prove a noncommutative analogue of Artin's solution to Hilbert's 17th problem, characterizing totally positive elements in terms of weighted sums of hermitian squares. As a consequence we obtain an earlier result of Procesi and Schacher and give a complete answer to their question about representation of elements as sums of hermitian squares. .

Mathematics Subject Classification (2010): 16K20, 11E39, 13J30.

Keywords and Phrases: Central simple algebra, involution, formally real field, hermitian form, signature, positivity, sum of hermitian squares.

Full text, 22p.: pdf 175k.

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