Remarks on One Combinatorial Application of the Aleksandrov–fenchel Inequalities

In 1981, Stanley applied the Aleksandrov–Fenchel Inequalities to prove a logarithmic concavity theorem for regular matroids. Using ideas from electrical network theory we prove a generalization of this for the wider class of matroids with the “half–plane property”. Then we explore a nest of inequalities for weighted basis–generating polynomials that are related to these ideas. As a first result from this investigation we find that every matroid of rank three or corank three satisfies a condition only slightly weaker than the conclusion of Stanley’s theorem. Dedicated with great admiration to Richard Stanley on the occasion of his 60th birthday.