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arXiv · 1811.01600

Log-Concave Polynomials III: Mason's Ultra-Log-Concavity Conjecture for Independent Sets of Matroids

Nima Anari·Kuikui Liu·Shayan Oveis Gharan·Cynthia Vinzant
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Abstract

We give a self-contained proof of the strongest version of Mason's conjecture, namely that for any matroid the sequence of the number of independent sets of given sizes is ultra log-concave. To do this, we introduce a class of polynomials, called completely log-concave polynomials, whose bivariate restrictions have ultra log-concave coefficients. At the heart of our proof we show that for any matroid, the homogenization of the generating polynomial of its independent sets is completely log-concave.

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Nima Anari, Kuikui Liu, Shayan Oveis Gharan, Cynthia Vinzant. 2018-11-05. Log-Concave Polynomials III: Mason's Ultra-Log-Concavity Conjecture for Independent Sets of Matroids. https://arxiv.org/abs/1811.01600

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