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Creators/Authors contains: "Penaguiao, Raul"

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  1. ; ; (, SIAM Journal on Discrete Mathematics)
    We prove that the number of tropical critical points of an affine matroid (M,e) is equal to the beta invariant of M. Motivated by the computation of maximum likelihood degrees, this number 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)*, where e is an element of M that is neither a loop nor a coloop. Equivalently, for a generic weight vector w on 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. This answers a question of Sturmfels. 
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  2. ; ; (, Séminaire lotharingien de combinatoire)
    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. 
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