Likelihood Degenerations

TL;DR

This paper introduces degeneration techniques inspired by physics to analyze the maximum likelihood degree of monomial critical points on algebraic varieties, connecting geometry, combinatorics, and algebra.

Contribution

It develops new degeneration methods for studying ML degrees, addressing questions in physics and algebraic geometry with theoretical and practical insights.

Findings

Degeneration techniques relate ML degree to geometric and combinatorial structures.

Bounded regions in discriminantal arrangements are characterized using these methods.

Connections between complex geometry, tropical combinatorics, and numerical algebra are established.

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.