Conference abstracts

Session A2 - Computational Algebraic Geometry

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Imaginary Projections of (Homogeneous) Polynomials

Thorsten Jörgens

Goethe University Frankfurt, Germany   -   joergens@math.uni-frankfurt.de

We introduce the imaginary projection of a general multivariate polynomial $f\in\mathbb{C}[\mathbf{z}]$ as the projection of the variety of $f$ onto its imaginary part, $\mathcal{I}(f) \ = \ \{\text{Im}(\mathbf{z}) \, : \, \mathbf{z} \in \mathcal{V}(f) \}$. 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. As a first fundamental geometric property we show that the connected components of the complement of the imaginary projections are convex.

For homogeneous (and hyperbolic) polynomials, we study the relation between imaginary projections and hyperbolicity cones. Hyperbolic polynomials became of interest in hyperbolic programming, where the set of feasible solutions is a hyperbolicity cone of a hyperbolic polynomial. It is a natural generalization of semidefinite programming.

Building upon this, for the homogeneous case we characterize the boundary of imaginary projections and we discuss a connection between hyperbolicity cones and certain components in the complement of the imaginary projection of inhomogeneous polynomials.

Joint work with Thorsten Theobald (Goethe University Frankfurt, Germany) and Timo de Wolff (Texas A&M University, USA).

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