The tropical Poincaré-Hopf theorem
arXiv:2007.11642 · doi:10.1016/j.jcta.2023.105733
Abstract
We express the beta invariant of a loopless matroid as tropical self-intersection number of the diagonal of its matroid fan (a "local" Poincaré-Hopf theorem). This provides another example of uncovering the "geometry" of matroids by expressing their invariants in terms of tropicalised geometric constructions. We also prove a global Poincaré-Hopf theorem and initiate the study of a more general tropical Lefschetz-Hopf trace formula by proving the two special cases of tropical curves and tropical tori.
31 pages, 3 figures; several improvements and corrections in the new version