Log-concavity of characteristic polynomials and the Bergman fan of matroids

Details Log-concavity of characteristic polynomials and the Bergman fan of matroids Log-concavity of characteristic polynomials and the Bergman fan of matroids

2011-04-13

arxiv.org/abs/1104.2519v2

Mathematics

Combinatorics

12 pages

Scientific paper

In a recent paper, the first author proved the log-concavity of the coefficients of the characteristic polynomial of a matroid realizable over a field of characteristic 0, answering a long-standing conjecture of Read in graph theory. We extend the proof to all realizable matroids, making progress towards a more general conjecture of Rota-Heron-Welsh. Our proof follows from an identification of the coefficients of the reduced characteristic polynomial as answers to particular intersection problems on a toric variety. The log-concavity then follows from an inequality of Hodge type.

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