A positivstellensatz for non-commutative polynomials

Authors:
J. William Helton and Scott A. McCullough

Translated by:

Journal:
Trans. Amer. Math. Soc. **356** (2004), 3721-3737

MSC (2000):
Primary 47A13

DOI:
https://doi.org/10.1090/S0002-9947-04-03433-6

Published electronically:
March 23, 2004

MathSciNet review:
2055751

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

Email:
helton@osiris.ucsd.edu

**Scott A. McCullough**

Affiliation:
Department of Mathematics, University of Florida, Gainesville, Florida 32611-8105

Email:
sam@math.ufl.edu

DOI:
https://doi.org/10.1090/S0002-9947-04-03433-6

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

Article copyright:
© Copyright 2004
American Mathematical Society