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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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A positivstellensatz for non-commutative polynomials
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by J. William Helton and Scott A. McCullough PDF
Trans. Amer. Math. Soc. 356 (2004), 3721-3737 Request permission

Abstract:

A non-commutative polynomial which is positive on a bounded semi-algebraic set of operators has a weighted sum of squares representation. This Positivstellensatz parallels similar results in the commutative case. A broader issue is, to what extent does real semi-algebraic geometry extend to non-commutative polynomials? Our “strict" Positivstellensatz is positive news, on the opposite extreme from strict positivity would be a Real Nullstellensatz. We give an example which shows that there is no non-commutative Real Nullstellensatz along certain lines. However, we include a successful type of non-commutative Nullstellensatz proved by George Bergman.
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Additional Information
  • J. William Helton
  • Affiliation: Department of Mathematics, University of California, San Diego, California 92093
  • MR Author ID: 84075
  • Email: helton@osiris.ucsd.edu
  • Scott A. McCullough
  • Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611-8105
  • MR Author ID: 220198
  • Email: sam@math.ufl.edu
  • Received by editor(s): January 6, 2003
  • Received by editor(s) in revised form: June 5, 2003
  • Published electronically: March 23, 2004
  • Additional Notes: The first author was partially supported by the the NSF, DARPA and Ford Motor Co.
    The second author was partially supported by NSF grant DMS-0140112
  • © Copyright 2004 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 356 (2004), 3721-3737
  • MSC (2000): Primary 47A13
  • DOI: https://doi.org/10.1090/S0002-9947-04-03433-6
  • MathSciNet review: 2055751