Hyperbolicity cones of elementary symmetric polynomials are spectrahedral

TL;DR

This paper proves that the hyperbolicity cones of elementary symmetric polynomials are spectrahedral, meaning they can be represented as slices of the positive semidefinite cone, using the matrix-tree theorem.

Contribution

It establishes the spectrahedral nature of hyperbolicity cones for elementary symmetric polynomials, connecting hyperbolic and semidefinite programming.

Findings

Hyperbolicity cones of elementary symmetric polynomials are spectrahedral.

The proof utilizes the matrix-tree theorem.

This result links hyperbolic polynomials with semidefinite representations.

Abstract

We prove that the hyperbolicity cones of elementary symmetric polynomials are spectrahedral, i.e., they are slices of the cone of positive semidefinite matrices. The proof uses the matrix--tree theorem, an idea already present in Choe et al.