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Remarks on one combinatorial application of the Aleksandrov-Fenchel inequalities

David G. Wagner
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Abstract

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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David G. Wagner. 2004-07-02. Remarks on one combinatorial application of the Aleksandrov-Fenchel inequalities. https://arxiv.org/abs/math/0406339

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