A criterion for the half-plane property

We establish a convenient necessary and sufficient condition for a multiaffine real polynomial to be stable, and use it to verify that the half-plane property holds for seven small matroids that resisted the efforts of Choe, Oxley, Sokal, and Wagner [5]. In recent years, matroid theory has found connections with certain analytic properties of real multivariate polynomials. These properties are abstractions of physical characteristics of an electrical network. Not all matroids exhibit the same physically sensible behaviour that graphs do. It is an interesting (and often challenging) problem to determine whether a given matroid satisfies one or another of these physicallymotivated conditions. In this paper we deduce a convenient necessary and sufficient criterion (Theorem 3(c)) for the “strong Rayleigh property”, and use it to verify this property for some small matroids, among them the Vámos cube V8. This supplements Brändén’s result [2] that the strong Rayleigh property is equivalent to the “half-plane property”, and resolves some questions left open by Choe, Oxley, Sokal, and Wagner [5]. Let Z(y1, ..., ym) be a polynomial with real coefficients, and let E = {1, ...,m}. The polynomial Z has the half-plane property (HPP) or is Hurwitz stable provided that whenever Re(ye) > 0 for all e ∈ E then Z(y1, ..., ym) 6= 0. The polynomial Z is stable provided that whenever Im(ye) > 0 for all e ∈ E then Z(y1, ..., ym) 6= 0. Note that a homogeneous polynomial Z is stable if and only if it is Hurwitz stable. If every variable ye for e ∈ E occurs in Z to at most the first power, then the polynomial Z is multiaffine. For any index e ∈ E let