To any [matroid][1] $M$ on ground set $E$ we associate the *characteristic polynomial*
$$p_M(t) = \sum_{A \subseteq E} (-1)^{|A|}t^{r(E) - r(S)}$$
where $r(A)$ denotes the *rank* of $A$.  Let $r(E) = r.$ It is known that
$$p_M(t) = a_0 t^r - a_1 t^{r-1} + \cdots + (-1)^r a_r$$
for $a_i \geq 0$ (i.e. coefficients alternate in sign). Moreover the coefficients are known to have some combinatorial interpretation. It was a long standing conjecture of Rota-Heron-Welsh that the sequence $(a_0, a_1, \cdots, a_r)$ is [log-concave][2].

In [Log-concavity of characteristic polynomials and the Bergman fan of matroids][3], Huh and Katz show that the Rota-Heron-Welsh conjecture is true for "realizable" matroids by making use of intersection theory and toric varieties. Adiprasito, Huh, and Katz prove the conjecture for general matroids in [Hodge Theory for Combinatorial Geometries][4].


  [1]: https://en.wikipedia.org/wiki/Matroid
  [2]: https://en.wikipedia.org/wiki/Logarithmically_concave_sequence
  [3]: https://arxiv.org/abs/1104.2519
  [4]: https://arxiv.org/abs/1511.02888