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trace-positive non-commutative polynomials

Author: Ronan Quarez
Journal: Proc. Amer. Math. Soc. 143 (2015), 3357-3370
MSC (2010): Primary 14P99, 15A63
Published electronically: February 13, 2015
MathSciNet review: 3348778
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Abstract: We give some examples of trace-positive non-commutative polynomials of degree $ 4$ in $ 3$ variables which are not cyclically equivalent to a sum of hermitian squares. Since some similar examples of degree $ 6$ in $ 2$ variables were alreay known, this settles a perfect analogy to Hilbert's result from the commutative context which says that positive (commutative) polynomials of degree $ d$ in $ n$ variables are not necessarily sums of squares, the first non-trivial cases being obtained for $ (d,n)=(4,3)$ and $ (d,n)=(6,2)$.

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Ronan Quarez
Affiliation: IRMAR (CNRS, URA 305), Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France

Keywords: Positive semidefinite, sums of squares, sums of hermitian squares, trace-positivity
Received by editor(s): May 14, 2013
Received by editor(s) in revised form: November 13, 2013, and January 9, 2014
Published electronically: February 13, 2015
Additional Notes: The first author was supported by French National Research Agency (ANR) project GEOLMI - Geometry and Algebra of Linear Matrix Inequalities with Systems Control Applications
Communicated by: Harm Derksen
Article copyright: © Copyright 2015 American Mathematical Society

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