Abstract
The generalized Lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices. We prove the conjecture for a multivariate generalization of the matching polynomial. This is further extended (albeit in a weaker sense) to a multivariate version of the independence polynomial for simplicial graphs. As an application, we give a new proof of the conjecture for elementary symmetric polynomials (originally due to Brändén). Finally, we consider a hyperbolic convolution of determinant polynomials generalizing an identity of Godsil and Gutman.
Highlights
A homogeneous polynomial h(x) ∈ R[x1, . . . , xn] is hyperbolic with respect to a vector e ∈ Rn if h(e) = 0, and if for all x ∈ Rn the univariate polynomial t → h(te−x) has only real zeros
We prove Conjecture 1.6 for a multivariate generalization of the matching polynomial (Theorem 2.16). We show that this implies Conjecture 1.6 for elementary symmetric polynomials (Theorem 2.19)
As an application of Theorem 2.16, we show that several wellknown instances of hyperbolic polynomials have spectrahedral hyperbolicity cones by realizing them as factors of the multivariate matching polynomial of Sn under some linear change of variables
Summary
Det(X ) is a hyperbolic polynomial with respect to I , and its hyperbolicity cone is the cone of positive semidefinite matrices. Hyperbolic and stable polynomials are related as follows, see [3, Prop. An) → det(x1 A1 + · · · xn An) of a vector space of the same dimension), whereas the set of hyperbolic polynomials of degree d on Rn has non-empty interior in the space of homogeneous polynomials of degree d in n variables (see [29]) and has the same dimension n +d −1 d. En) is spectrahedral if there is a homogeneous polynomial q(x) and real symmetric matrices A1, . 2, we prove Conjecture 1.6 for a multivariate generalization of the matching polynomial (Theorem 2.16).
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