A Note on the Hyperbolicity Cone of the Specialized V\'amos Polynomial

TL;DR

This paper investigates the hyperbolicity cone of a specialized Vamos polynomial, demonstrating through heuristic methods that it forms a spectrahedron, despite the polynomial's powers lacking definite determinantal representations.

Contribution

It proves that the hyperbolicity cone of the specialized Vamos polynomial is a spectrahedron using heuristic techniques, advancing understanding of hyperbolic polynomials and their cones.

Findings

The specialized Vamos polynomial is hyperbolic of degree four.

Its powers do not admit definite determinantal representations.

The hyperbolicity cone is a spectrahedron.

Abstract

The specialized V\'amos polynomial is a hyperbolic polynomial of degree four in four variables with the property that none of its powers admits a definite determinantal representation. We will use a heuristical method to prove that its hyperbolicity cone is a spectrahedron.