Quasi-convex free polynomials
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- by S. Balasubramanian and S. McCullough PDF
- Proc. Amer. Math. Soc. 142 (2014), 2581-2591 Request permission
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.References
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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
- Email: bsriram@iiserkol.ac.in
- S. McCullough
- Affiliation: Department of Mathematics, The University of Florida, Box 118105, Gainesville, Florida 32611-8105
- MR Author ID: 220198
- Email: sam@ufl.edu
- 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
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 2581-2591
- MSC (2010): Primary 15A24, 47A63, 08B20
- DOI: https://doi.org/10.1090/S0002-9939-2014-11984-8
- MathSciNet review: 3209314