# Hyperbolicity cones and imaginary projections

## TL;DR

This paper explores the relationship between imaginary projections of polynomials and hyperbolicity cones, providing bounds on the number of components and demonstrating the potential for many convex bounded components.

## Contribution

It establishes a connection between imaginary projections and hyperbolicity cones, offering bounds and showing the possibility of many convex components for polynomials.

## Key findings

- Upper bound for the number of components in the complement of imaginary projections
- Polynomials can have arbitrarily many convex bounded components in their imaginary projection complement
- Relation between imaginary projections and hyperbolicity cones for homogeneous polynomials

## 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, $\mathcal{I}(f) \ = \ \{\text{Im}(\mathbf{z}) \, : \, \mathbf{z} \in \mathcal{V}(f) \}$. 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. 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 $\mathcal{I}(f)^{c}$ 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 $\mathcal{I}(f)^{c}$.