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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