Remarks on one combinatorial application of the Aleksandrov-Fenchel inequalities

Details Remarks on one combinatorial application of the Aleksandrov-Fenchel inequalities Remarks on one combinatorial application of the Aleksandrov-Fenchel inequalities

2004-06-17

arxiv.org/abs/math/0406339v2

Mathematics

Combinatorics

18 pages, one figure, two tables. Minor typos and references fixed

Scientific paper

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.

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