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Log-concavity of characteristic polynomials and the Bergman fan of matroids. (English) Zbl 1258.05021
Summary: In a recent paper, the first author [J. Am. Math. Soc. 25, No. 3, 907–927 (2012; Zbl 1243.14005)] 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.

MSC:
05B35 Combinatorial aspects of matroids and geometric lattices
14B05 Singularities in algebraic geometry
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