  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.
