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


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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Correspondence to Eric Katz.

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Huh, J., Katz, E. Log-concavity of characteristic polynomials and the Bergman fan of matroids. Math. Ann. 354, 1103–1116 (2012).

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  • Characteristic Polynomial
  • Toric Variety
  • Hyperplane Arrangement
  • Cartier Divisor
  • Chromatic Polynomial