Polar Degrees of Matroids

Technical Summary: Polar Degrees of Matroids

Problem Statement
This paper investigates the geometric properties of reciprocal linear spaces, specifically the closure $L^{-1}$ of the image of a linear subspace $L \subseteq \mathbb{P}^n$ under the coordinate-wise inverse map (the Cremona transformation). The central object of study is the sequence of polar degrees of $L^{-1}$, which encode the degrees of its polar varieties and, crucially, the degree of its projective dual variety $(L^{-1})^\vee$.

The work is motivated by recent conjectures by Matsubara-Heo and Telen regarding the principal matroid determinant $E_L$. These conjectures posited that:

1. $L^{-1}$ is dual defective (i.e., $\dim((L^{-1})^\vee) < n-1$) if and only if the associated matroid $M(L)$ is disconnected.
2. If $M(L)$ is connected, the degree of the dual variety is $2^d \beta(M)$, where $d = \dim(L)$ and $\beta(M)$ is the beta invariant of the matroid.
3. The factorization of $E_L$ involves specific multiplicities related to local multiplicities of $L^{-2}$ along strata.

While the third conjecture regarding multiplicities remains open (addressed in future work [BKW26]), this paper provides a complete combinatorial description of the polar degrees, thereby settling the first two conjectures and offering a unified formula for all polar degrees.

Methodology
The authors employ a synthesis of algebraic geometry, tropical geometry, and matroid theory. The core strategy involves translating the geometric problem of computing intersection numbers on the conormal variety into a combinatorial problem on the bipermutohedral fan.

1. Conormal Varieties and Parametrization: The study centers on the conormal variety $\text{Con}(L^{-1}) \subseteq \mathbb{P}^n \times (\mathbb{P}^n)^*$. The authors utilize a rational parametrization $\psi: L \times L^\perp \dashrightarrow \mathbb{P}^n \times (\mathbb{P}^n)^*$, $(x, y) \mapsto (x^{-1}, x^2 y)$, which maps to the conormal variety.
2. Resolution of Indeterminacy: To compute degrees, the rational map must be resolved. The authors resolve the base locus by passing from a product of projective spaces to the bipermutohedral variety $X(\Sigma_{E,E})$, a smooth projective toric variety introduced by Adiprasito, Huh, and Katz [ADH22]. This variety hosts the conormal fan $\Sigma_{M, M^\perp}$ of the matroid $M$.
3. Tropicalization and Weight Tracking: The authors use tropical geometry to relate the polar degrees of $L^{-1}$ to the multidegree of a simpler map $\phi(x, y) = (x^{-1}, xy)$. By analyzing the tropicalization of the Hadamard product $L \star L^\perp$ and tracking the weights of cones under finite maps (specifically squaring maps), they extract factors of $2$ and $m$ (for generalized powers) from the intersection numbers.
4. Combinatorial Reduction: The intersection numbers on the conormal variety are expressed as sums over decreasing flags of flats in the matroid $M$. These sums are then related to the coefficients of the reduced characteristic polynomial $\chi_M(t)$ via a recursive argument involving beta invariants of minors.

Key Contributions and Results

• Main Theorem (Theorem A): The paper establishes a closed formula for the polar degrees $\mu_j(L^{-1})$ of the reciprocal linear space. Let $\chi_M(t)$ be the reduced characteristic polynomial of $M$. The generating function for the polar degrees is given by:
  $$\sum_{j=0}^d \mu_j(L^{-1}) t^j = (-2t - 1)^d \chi_M\left( \frac{2t}{2t + 1} \right)$$
  Equivalently, for the dual degrees $\delta_i(L^{-1}) = \mu_{d-i}(L^{-1})$:
  $$\sum_{i=0}^d \delta_i(L^{-1}) t^i = (-t - 2)^d \chi_M\left( \frac{2}{t + 2} \right)$$

• Resolution of Conjectures:

  • Conjecture 1.1 (Dual Defectivity): The formula confirms that $L^{-1}$ is dual defective if and only if $\beta(M) = 0$, which occurs precisely when $M$ is disconnected.
  • Conjecture 1.2 (Dual Degree): For connected matroids, the dual degree is explicitly $\deg((L^{-1})^\vee) = 2^d \beta(M)$.

• Generalizations:

  • Higher Powers: The results are extended to the reciprocal linear space $L^{-m}$ (image under coordinate-wise $m$-th power inverse). Theorem 4.8 provides a formula involving $m$ and the number of connected components $c(M)$:
    $$\sum_{j=0}^d \mu_j(L^{-m}) t^j = m^{1-c(M)} (-(m+1)t - m)^d \chi_M\left( \frac{(m+1)t}{(m+1)t + m} \right)$$
  • Chern-Mather Classes: The paper derives a combinatorial formula for the degrees of the Chern-Mather classes of $L^{-1}$ in terms of $\chi_M$.
  • Distance Degrees: The generic distance degree of $L^{-1}$ is shown to be $(-3)^d \chi_M(2/3)$.

Significance
The paper demonstrates that the polar degrees of reciprocal linear spaces are entirely determined by the combinatorial invariants of the underlying matroid. By expressing these geometric invariants as coefficients of a substitution of the reduced characteristic polynomial, the authors bridge the gap between the geometry of conormal varieties and the combinatorics of the bipermutohedral fan.

The work settles two open conjectures regarding the structure of the principal matroid determinant $E_L$, clarifying which factors appear and their degrees. Furthermore, it reinforces the utility of tropical methods and the machinery developed in [ADH22] for resolving long-standing problems in matroid theory and algebraic statistics. The authors note that while the log-concavity of the coefficients of the resulting polynomial follows from the log-concavity of $\chi_M$ (proven in [AHK18]), the explicit combinatorial formula provides a direct geometric interpretation of these coefficients as intersection numbers.