Euler Stratifications of Hypersurface Families

TL;DR

This paper introduces algorithms to stratify hypersurface families based on their Euler characteristic, linking topology to applications in physics and statistics, and exploiting critical point computations for very affine hypersurfaces.

Contribution

It presents new algebro-geometric algorithms for Euler stratifications and explores their relation to critical points in very affine hypersurfaces.

Findings

Algorithms compute all Euler strata efficiently.

Euler stratifications relate to the number of master integrals.

They describe the maximum likelihood degree dependence.

Abstract

We stratify families of projective and very affine hypersurfaces according to their topological Euler characteristic. Our new algorithms compute all strata using algebro-geometric techniques. For very affine hypersurfaces, we investigate and exploit the relation to critical point computations. Euler stratifications are relevant in particle physics and algebraic statistics. They fully describe the dependence of the number of master integrals, respectively the maximum likelihood degree, on kinematic or model parameters.