The tropical critical points of an affine matroid

TL;DR

This paper establishes that the number of tropical critical points of an affine matroid equals its beta invariant, linking combinatorial invariants with geometric intersections related to maximum likelihood degrees.

Contribution

It proves a new equality between tropical critical points count and the beta invariant of an affine matroid, answering a question posed by Sturmfels.

Findings

Number of tropical critical points equals the beta invariant of the matroid.

Provides a geometric interpretation involving Bergman fans and maximum likelihood degrees.

Connects combinatorial invariants with tropical geometry and algebraic statistics.

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.