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 𝒜 such that Mag(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.
Related
- On the monotonicity of magnitude functions of negative type finite semimetric spaces
- Representation zeta functions of self-similar branched groups
- Representation zeta functions of wreath products with finite groups
- Chromatic word-quasisymmetric functions of matroids
- Zeta functions of quasi-split reductive algebraic groups over local fields
BibTeX
@misc{koizumi2026magnitude,
title = {{Magnitude and motivic zeta functions of matroids}},
author = {Junnosuke Koizumi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07977},
eprint = {2609.07977},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07977},
}
RIS
TY - UNPB
TI - Magnitude and motivic zeta functions of matroids
AU - Junnosuke Koizumi
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07977
UR - https://arxiv.org/abs/2609.07977
AB - 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 𝒜 such that Mag(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.
ER -