A real stable extension of the Vamos matroid polynomial

Abstract

In 2004, Choe, Oxley, Sokal and Wagner established a tight connection between matroids and multiaffine real stable polynomials. Recently, Brändén used this theory and a polynomial coming from the Vámos matroid to disprove the generalized Lax conjecture.Here we present a 10-element extension of the Vámos matroid and prove that its basis generating polynomial is real stable (i.e. that the matroid has the half-plane property). We do this via large sums of squares computations and a criterion for real stability given by Wagner and Wei. Like the Vámos matroid, this matroid is not representable over any field and no power of its basis generating polynomial can be written as the determinant of a linear matrix with positive semidefinite Hermitian forms.

Matroids on Eight Elements with the Half-plane Property and Related Concepts

“…In a celebrated article Brändén [5] disproved a predecessor of the generalized Lax Conjecture by showing that the Vámos matroid V 8 is not weakly determinantal although it has the half-plane property [29]. Further such matroids were constructed in [1,7]. In this section we will add some further examples to this list.…”

“…Further such examples were given in [7,1] and will be given in Section 5.1 of this article (on the other hand, see [18] for some positive result). This indicates that bases generating polynomials of matroids might be a good testing ground for the (current version of the) generalized Lax conjecture as well.…”

A Note on the Hyperbolicity Cone of the Specialized Vámos Polynomial

“…It is easy to see, that both properties spread to h 4 . All polynomials that are known to have these properties are constructed in some way from the Vámos matroid, see also [3].…”

“…In Section 3, we will apply this method to the specialized Vámos polynomial h 4 , which is a hyperbolic polynomial of degree four in four variables, with the property that no power h N 4 has a definite determinantal representation. Note that only very few polynomials [1,3] are known to have this property and all of these come from the Vámos matroid. The polynomial h 4 and its relatives frequently serve as counterexamples for questions concerning hyperbolic polynomials [1,11,13] and thus it is natural to check whether it might also be a counterexample to the Generalized Lax Conjecture.…”

Determinantal representations and Bézoutians

“…Actually Brändén [3] found a hyperbolic polynomial h in four variables such that no power h N admits a definite determinantal representation. More about these topics can be found in [5,14,15,20,23].…”