# On the connectivity of the hyperbolicity region of irreducible polynomials

## TL;DR

This paper provides a simple proof demonstrating that an irreducible hyperbolic polynomial possesses a unique pair of hyperbolicity cones, clarifying the structure of these polynomials.

## Contribution

It offers an elementary proof establishing the connectivity of the hyperbolicity region for irreducible hyperbolic polynomials, a previously less straightforward result.

## Key findings

- Irreducible hyperbolic polynomials have a single pair of hyperbolicity cones.
- The proof simplifies understanding of the hyperbolicity region's structure.
- The result confirms the connectedness of the hyperbolicity region for such polynomials.

## Abstract

We give an elementary proof for the fact that an irreducible hyperbolic polynomial has only one pair of hyperbolicity cones.