Intersection theory of matroids: variations on a theme

Chow rings of toric varieties, which originate in intersection theory, feature a rich combinatorial structure of independent interest. We survey four different ways of computing in these rings, due to Billera, Brion, Fulton--Sturmfels, and Allermann--Rau. We illustrate the beauty and power of these methods by giving four proofs of Huh and Huh--Katz's formula $\mu^k(M) = deg_M(\alpha^{r-k} \beta^k)$ for the coefficients of the reduced characteristic polynomial of a matroid $M$ as the mixed intersection numbers of the hyperplane and reciprocal hyperplane classes $\alpha$ and $\beta$ in the Chow ring of $M$. Each of these proofs sheds light on a different aspect of matroid combinatorics, and provides a framework for further developments in the intersection theory of matroids. Our presentation is combinatorial, and does not assume previous knowledge of toric varieties, Chow rings, or intersection theory.