Log-concavity of characteristic polynomials and the Bergman fan of matroids |
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Authors: | June Huh Eric Katz |
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Institution: | 1. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA 2. Department of Combinatorics and Optimization, University of Waterloo, Waterloo, ON, N2L 3G1, Canada
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Abstract: | 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?CHeron?CWelsh. 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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