Federico Ardila-Mantilla, Christopher Eur, Raul Penaguiao
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SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1930-1942, June 2024. Abstract. We prove that the number of tropical critical points of an affine matroid [math] is equal to the beta invariant of [math]. Motivated by the computation of maximum likelihood degrees, this number is defined to be the degree of the intersection of the Bergman fan of [math] and the inverted Bergman fan of [math], where [math] is an element of [math] that is neither a loop nor a coloop. Equivalently, for a generic weight vector [math] on [math], this is the number of ways to find weights [math] on [math] and [math] on [math] with [math] such that, on each circuit of [math] (resp., [math]), the minimum [math]-weight (resp., [math]-weight) occurs at least twice. This answers a question of Sturmfels.
期刊介绍:
SIAM Journal on Discrete Mathematics (SIDMA) publishes research papers of exceptional quality in pure and applied discrete mathematics, broadly interpreted. The journal''s focus is primarily theoretical rather than empirical, but the editors welcome papers that evolve from or have potential application to real-world problems. Submissions must be clearly written and make a significant contribution.
Topics include but are not limited to:
properties of and extremal problems for discrete structures
combinatorial optimization, including approximation algorithms
algebraic and enumerative combinatorics
coding and information theory
additive, analytic combinatorics and number theory
combinatorial matrix theory and spectral graph theory
design and analysis of algorithms for discrete structures
discrete problems in computational complexity
discrete and computational geometry
discrete methods in computational biology, and bioinformatics
probabilistic methods and randomized algorithms.