Intersection theory of matroids: variations on a theme

TL;DR

The paper connects the coefficients μ^k of the reduced characteristic polynomial of a matroid to degrees deg_M(α^{r-k}β^k) in the matroid Chow ring, unifying four combinatorial viewpoints on intersection theory in toric settings. It develops and compares four descriptions of the Chow ring of a matroid (quotient of a polynomial ring, piecewise-polynomial functions, Minkowski weights, and tropical intersection) and uses these to prove the central identity via four distinct arguments, all anchored in the rich geometry of toric varieties and matroid fans. Leveraging the four approaches, the work illuminates how matroid invariants arise from intersection-theoretic structures, and connects to broader themes such as Hodge–Riemann theory and log-concavity. The survey further outlines ongoing developments linking Tutte polynomials, beta-invariants, and valuative matroid invariants to Minkowski- and tropical-geometric frameworks, underscoring the deep interplay between combinatorics, geometry, and intersection theory in matroids.

Abstract

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 $μ^k(M) = deg_M(α^{r-k} β^k)$ for the coefficients of the reduced characteristic polynomial of a matroid $M$ as the mixed intersection numbers of the hyperplane and reciprocal hyperplane classes $α$ and $β$ 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.

Figures (12)

• Figure 1: A graph $G$ with $\chi_G(q) = q^4-4q^3 +5q^2- 2q.$ and $\mu^0=1, \mu^1=3, \mu^2=2$.
• Figure 2: The braid fan $\Sigma_{[0,2]}$.
• Figure 3: The Courant functions $x_0, x_1, x_2, x_{01}, x_{02}, x_{12}$ on $\Sigma_{[0,2]}$. Each function $x_S$ equals $1$ on the marked primitive ray $\mathop{\mathrm{\mathbf e}}\nolimits_S$ and $0$ on the others.
• Figure 4: A vector configuration, its lattice of flats, and its matroid fan.
• Figure :

Theorems & Definitions (23)

• Theorem 1.1
• Theorem 2.1
• Definition 2.2
• Example 2.3
• Example 2.4
• Example 2.5
• Example 2.6
• Example 2.7
• Example 2.8
• Example 2.9