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 [Y.-B. Choe, J.G. Oxley, A.D. Sokal, D.G. Wagner, Homogeneous polynomials with the half-plane property, Adv. Appl. Math. 32 (2004) 88–187].