On the geometry of flag Hilbert-Poincaré series for matroids
Lukas Kühne, Joshua Maglione
TL;DR
This work extends the coarse flag Hilbert--Poincaré series to matroids and leverages oriented-matroid topes to study the resulting polynomials. A central result shows that for orientable matroids of rank $r$, the numerator at $Y=1$ satisfies $E_r^{\mathsf{A}}(T) \leq \dfrac{\mathcal{N}_M(1,T)}{\pi_M(1)}$, with equality iff the matroid is simplicial, and reveals a symmetry $\mathcal{N}_M(1,T^{-1})=T^{r-1}\mathcal{N}_M(1,T)$. The authors propose conjectured bounds $(1+T)^{r-1} < \dfrac{\mathcal{N}_M(1,T)}{\pi_M(1)} < E_r^{\mathsf{B}}(T)$, confirm them in low rank cases, and discuss real-rootedness and palindromicity properties. By studying extremal families—uniform matroids for the upper bound and finite projective geometries for the lower bound—the paper situates the coarse flag polynomial within a broader valuative and geometric framework, linking to motivic zeta functions and the $cd$-index of topes. Overall, the work connects combinatorial flag invariants with geometric-topological structures of tope lattices, offering practical criteria for non-orientability and new avenues for positivity and real-rootedness questions in matroid theory.
Abstract
We extend the definition of coarse flag Hilbert--Poincaré series to matroids; these series arise in the context of local Igusa zeta functions associated to hyperplane arrangements. We study these series in the case of oriented matroids by applying geometric and combinatorial tools related to their topes. In this case, we prove that the numerators of these series are coefficient-wise bounded below by the Eulerian polynomial and equality holds if and only if all topes are simplicial. Moreover this yields a sufficient criterion for non-orientability of matroids of arbitrary rank.