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Quasi-convex free polynomials

Authors: S. Balasubramanian and S. McCullough
Journal: Proc. Amer. Math. Soc. 142 (2014), 2581-2591
MSC (2010): Primary 15A24, 47A63, 08B20
Published electronically: May 13, 2014
MathSciNet review: 3209314
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Abstract: Let $ \mathbb{R}\langle x \rangle $ denote the ring of polynomials in $ g$ freely noncommuting variables $ x=(x_1,\dots ,x_g)$. There is a natural involution $ *$ on $ \mathbb{R}\langle x \rangle $ determined by $ x_j^*=x_j$ and $ (pq)^*=q^* p^*$, and a free polynomial $ p\in \mathbb{R}\langle x \rangle $ is symmetric if it is invariant under this involution. If $ X=(X_1,\dots ,X_g)$ is a $ g$ tuple of symmetric $ n\times n$ matrices, then the evaluation $ p(X)$ is naturally defined and further $ p^*(X)=p(X)^*$. In particular, if $ p$ is symmetric, then $ p(X)^*=p(X)$. The main result of this article says if $ p$ is symmetric, $ p(0)=0$ and for each $ n$ and each symmetric positive definite $ n\times n$ matrix $ A$ the set $ \{X:A-p(X)\succ 0\}$ is convex, then $ p$ has degree at most two and is itself convex, or $ -p$ is a hermitian sum of squares.

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Additional Information

S. Balasubramanian
Affiliation: Department of Mathematics and Statistics, Indian Institute of Science Education and Research (IISER) – Kolkata, Mohanpur Campus, Nadia District, Pin: 741246, West Bengal, India

S. McCullough
Affiliation: Department of Mathematics, The University of Florida, Box 118105, Gainesville, Florida 32611-8105

Keywords: Free polynomials, quasi-convex, free real algebraic geometry
Received by editor(s): March 6, 2012
Received by editor(s) in revised form: August 11, 2012
Published electronically: May 13, 2014
Additional Notes: The research of the second author was supported by NSF grants DMS 0758306 and 1101137
Communicated by: Richard Rochberg
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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