Remarks on one combinatorial application of the Aleksandrov-Fenchel inequalities
Abstract
Description
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.
18 pages, one figure, two tables. Minor typos and references fixed
18 pages, one figure, two tables. Minor typos and references fixed