# Likelihood degenerations

Abstract: Computing all critical points of a monomial on a very affine variety is a fundamental task in algebraic statistics, particle physics and other fields. The number of critical points is known as the maximum likelihood (ML) degree. When the variety is smooth, it coincides with the Euler characteristic. We introduce degeneration techniques that are inspired by the soft limits in CEGM theory, and we answer several questions raised in the physics literature. These pertain to bounded regions in discriminantal arrangements and to moduli spaces of point configurations. We present theory and practise, connecting complex geometry, tropical combinatorics, and numerical nonlinear algebra.

In the following we showcase two computations discussed in the paper in Jupyter notebooks. The first presents the computation of ML degrees of matroid strata:

In the second notebook we compute the tropical solutions to the soft limit scattering equations as explained in Section 8 of the paper.

These notebooks together with the underlying code and the matroid database can be downloaded `here`

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The files concerning the ML degrees of small matroids and their certification are available on Zenodo.

Project page created: 22/07/2021

Project contributors: Daniele Agostini, Taylor Brysiewicz, Claudia Fevola, Lukas Kühne, Bernd Sturmfels, and Simon Telen

Corresponding author of this page: Lukas Kühne, lukas.kuhne@mis.mpg.de

Software used: Julia (Version 1.6.0)