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, Branden used this theory and a polynomial coming from the Vamos matroid to disprove the generalized Lax conjecture. Here we present a 10-element extension of the Vamos 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 Vamos 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.