The tropical Poincar\'e-Hopf theorem
Algebraic Geometry
2023-02-22 v2 Combinatorics
Abstract
We express the beta invariant of a loopless matroid as tropical self-intersection number of the diagonal of its matroid fan (a "local" Poincar\'e-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 Poincar\'e-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.
Keywords
Cite
@article{arxiv.2007.11642,
title = {The tropical Poincar\'e-Hopf theorem},
author = {Johannes Rau},
journal= {arXiv preprint arXiv:2007.11642},
year = {2023}
}
Comments
31 pages, 3 figures; several improvements and corrections in the new version