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arXiv · 2504.15518

The Ehrhart polynomial of a matroid specializes to the beta invariant

Anastasia Chavez·Galen Dorpalen-Barry·Luis Ferroni·Fu Liu·Felipe Rincón·Andrés R. Vindas-Meléndez
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Abstract

We show that the linear coefficient of the Ehrhart polynomial of a matroid base polytope evaluated at $t-1$ is equal to, up to normalization, the $β$-invariant of the matroid. This yields a lattice-point counting formula for the $β$-invariant and establishes a new and unexpected positivity property of Ehrhart polynomials of matroid polytopes.

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Anastasia Chavez, Galen Dorpalen-Barry, Luis Ferroni, Fu Liu, Felipe Rincón, Andrés R. Vindas-Meléndez. 2025-10-23. The Ehrhart polynomial of a matroid specializes to the beta invariant. https://arxiv.org/abs/2504.15518

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