Magnitude and motivic zeta functions of matroids

Abstract

We prove that the magnitude of the tope graph of a simple oriented matroid is the specialization at $x=-1$ of the motivic zeta function of its underlying matroid. This refines the Las Vergnas--Zaslavsky theorem and defines magnitude for arbitrary matroids. For simple orientable matroids, we compute the order of the pole at $q=-1$ from chains of flats and identify it with the Varchenko--Gelfand degree of the tope parity function. Comparing pole orders, we construct a rank-six real arrangement $\mathcal A$ such that $\operatorname{Mag}(\mathcal A;-t)$ has infinitely many negative coefficients, disproving Koizumi--Liu's eventual sign alternation conjecture. We then construct a canonical multiplicative Varchenko--Gelfand filtration on the mod-$2$ magnitude cohomology of a simple oriented matroid and show that its graded dimensions recover the full motivic zeta function.