To any [matroid][1] $M$ on ground set $E$ we associate the *characteristic polynomial* $$p_M(t) = \sum_{A \subseteq E} (-1)^{|A|}t^{r(E) - r(A)}$$ 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]. **Edit:** I have found a paper [Matroid theory for algebraic geometers][5] by Katz which gives a expository view of the work of Huh and Katz mentioned above as well as more background. It looks like a nice source which may be of interest to viewers of this question; so, I will share it here. [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 [5]: https://arxiv.org/abs/1409.3503