The tropical critical points of an affine matroid
Abstract
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.
Keywords
Cite
@article{arxiv.2212.08173,
title = {The tropical critical points of an affine matroid},
author = {Federico Ardila-Mantilla and Christopher Eur and Raul Penaguiao},
journal= {arXiv preprint arXiv:2212.08173},
year = {2024}
}
Comments
15 pages, 3 figures. v3: minor revisions. v2: section 4 has been rewritten, revising a flaw in v1. To appear in SIDMA