The maximum likelihood degree of a very affine variety

TL;DR

This paper establishes a deep connection between the maximum likelihood degree of smooth very affine varieties and their topological Euler characteristic, extending classical results from hyperplane arrangements to broader algebraic varieties.

Contribution

It generalizes the relationship between maximum likelihood degree and topological invariants to smooth very affine varieties, including a strengthened version involving Chern-Schwartz-MacPherson classes.

Findings

Maximum likelihood degree equals the signed topological Euler characteristic.

The result extends to relate critical points to Chern-Schwartz-MacPherson classes under generic conditions.

Recovers and generalizes classical theorems like the deletion-restriction formula and Kouchnirenko's theorem.

Abstract

We show that the maximum likelihood degree of a smooth very affine variety is equal to the signed topological Euler characteristic. This generalizes Orlik and Terao's solution to Varchenko's conjecture on complements of hyperplane arrangements to smooth very affine varieties. For very affine varieties satisfying a genericity condition at infinity, the result is further strengthened to relate the variety of critical points to the Chern-Schwartz-MacPherson class. The strengthened version recovers the geometric deletion-restriction formula of Denham et al. for arrangement complements, and generalizes Kouchnirenko's theorem on the Newton polytope for nondegenerate hypersurfaces.