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Séminaire Lotharingien de Combinatoire, 89B.28 (2023), 12 pp.

# Federico Ardila-Mantilla, Christopher Eur and
Raúl Penaguião

# The Tropical Critical Points of an Affine Matroid

**Abstract.**
We prove that the maximum likelihood degree of a matroid *M* equals its beta invariant β(*M*). For an element *e* of *M* that is neither a loop nor a coloop, this is defined to be the degree of the intersection of the Bergman fan of (*M*,*e*) and the inverted Bergman fan of *N* = (*M*/*e*)^{⊥}.
Equivalently, for a generic vector w ∈ **R**^{E-e}, this is the number of ways to find weights (0,x) on *M* and y on *N* with x+y = w such that on each circuit of *M* (resp. *N*), the minimum x-weight (resp. y-weight) occurs at least twice.

Received: November 15, 2022.
Accepted: February 20, 2023.
Final version: April 1, 2023.

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