To any matroid $M$ 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.
In Log-concavity of characteristic polynomials and the Bergman fan of matroids, 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.
