Hyperbolicity cones and imaginary projections
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- by Thorsten Jörgens and Thorsten Theobald PDF
- Proc. Amer. Math. Soc. 146 (2018), 4105-4116 Request permission
Abstract:
Recently, the authors and de Wolff introduced the imaginary projection of a polynomial $f\in \mathbb {C}[\mathbf {z}]$ as the projection of the variety of $f$ onto its imaginary part, $\mathcal {I}(f) \ = \ \{Im(\mathbf {z}) : \mathbf {z} \in \mathcal {V}(f) \}$. Since a polynomial $f$ is stable if and only if $\mathcal {I}(f) \cap \mathbb {R}_{>0}^n \ = \ \emptyset$, the notion offers a novel geometric view underlying stability questions of polynomials. In this article, we study the relation between the imaginary projections and hyperbolicity cones, where the latter ones are only defined for homogeneous polynomials. Building upon this, for homogeneous polynomials we provide a tight upper bound for the number of components in the complement $\mathcal {I}(f)^{\mathsf {c}}$ and thus for the number of hyperbolicity cones of $f$. And we show that for $n \ge 2$, a polynomial $f$ in $n$ variables can have an arbitrarily high number of strictly convex and bounded components in $\mathcal {I}(f)^{\mathsf {c}}$.References
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Additional Information
- Thorsten Jörgens
- Affiliation: Goethe-Universität, FB 12 – Institut für Mathematik, Postfach 11 19 32, D-60054 Frankfurt am Main, Germany
- Email: joergens@math.uni-frankfurt.de
- Thorsten Theobald
- Affiliation: Goethe-Universität, FB 12 – Institut für Mathematik, Postfach 11 19 32, D-60054 Frankfurt am Main, Germany
- MR Author ID: 618735
- ORCID: 0000-0002-5769-0917
- Email: theobald@math.uni-frankfurt.de
- Received by editor(s): August 14, 2017
- Received by editor(s) in revised form: January 4, 2018
- Published electronically: May 24, 2018
- Communicated by: Patricia Hersh
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 4105-4116
- MSC (2010): Primary 14P10, 12D10, 52A37
- DOI: https://doi.org/10.1090/proc/14081
- MathSciNet review: 3834642