Real Algebraic and Analytic Geometry |

Signatures of hermitian forms, positivity, and an answer to a question of Procesi and Schacher.

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Submission: 2016, September 27.

*Abstract:
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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