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Positivstellensätze for noncommutative rational expressions


Author: J. E. Pascoe
Journal: Proc. Amer. Math. Soc. 146 (2018), 933-937
MSC (2010): Primary 13J30, 16K40, 47L07; Secondary 15A22, 26C15, 47A63
DOI: https://doi.org/10.1090/proc/13773
Published electronically: September 14, 2017
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Abstract | References | Similar Articles | Additional Information

Abstract: We derive some Positivstellensätze for noncommutative rational expressions from the Positivstellensätze for noncommutative polynomials.
Specifically, we show that if a noncommutative rational expression is positive on a polynomially convex set, then there is an algebraic certificate witnessing that fact. As in the case of noncommutative polynomials, our results are nicer when we additionally assume positivity on a convex set, that is, we obtain a so-called ``perfect Positivstellensatz'' on convex sets.


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Additional Information

J. E. Pascoe
Affiliation: Department of Mathematics, Washington University in St. Louis, 1 Brookings Drive, Campus Box 1146, St. Louis, Missouri 63130
Email: pascoej@math.wustl.edu

DOI: https://doi.org/10.1090/proc/13773
Keywords: Noncommutative rational function, positive rational function, Hilbert's 17th problem, noncommutative Positivstellensatz
Received by editor(s): March 21, 2017
Received by editor(s) in revised form: April 12, 2017
Published electronically: September 14, 2017
Additional Notes: This research was supported by NSF Mathematical Science Postdoctoral Research Fellowship DMS 1606260.
Communicated by: Stephan Ramon Garcia
Article copyright: © Copyright 2017 American Mathematical Society

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