Polar Degrees of Matroids

We show that the polar degrees of the coordinate-wise inverse of a linear subspace $L\subseteq {\mathbb{P}}^n$ are given by the coefficients of a substitution of the reduced characteristic polynomial of the associated matroid $\mathrm{M}(L)$. Our proof connects the geometry of conormal varieties of reciprocal linear spaces to the combinatorial conormal fan of $\mathrm{M}(L)$. As a corollary, we settle two open conjectures regarding matroid discriminants.