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arXiv · 2007.11642

The tropical Poincar\'e-Hopf theorem

Johannes Rau
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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.

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Johannes Rau. 2020-07-22. The tropical Poincar\'e-Hopf theorem. https://doi.org/10.1016/j.jcta.2023.105733

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