Hypersurface Arrangements with Generic Hypersurfaces Added

TL;DR

This paper investigates the Euler characteristic of hypersurface arrangement complements with generic hypersurfaces, providing formulas, bounds, and an efficient sampling method for hyperplane arrangements.

Contribution

It introduces a new formula for the Euler characteristic involving Milnor numbers and degrees, and demonstrates an efficient sampling technique for hyperplane arrangement regions.

Findings

Euler characteristic depends on characteristic polynomial and degrees

Adding a degree-two hypersurface enables efficient interior point sampling

Provides a new upper bound on the number of regions in hypersurface arrangements

Abstract

The Euler characteristic of a very affine variety encodes the number of critical points of the likelihood equation on this variety. In this paper, we study the Euler characteristic of the complement of a hypersurface arrangement with generic hypersurfaces added. For hyperplane arrangements, it depends on the characteristic polynomial coefficients and generic hypersurface degrees. As a corollary, we show that adding a degree-two hypersurface to a real hyperplane arrangement enables efficient sampling of a single interior point from each region in the complement. We compare the method to existing alternatives and demonstrate its efficiency. For hypersurface arrangements, the Euler characteristic is expressed in terms of Milnor numbers and generic hypersurface degrees. This formulation further yields a novel upper bound on the number of regions in the complement of a hypersurface arrangement.