Hypersurface Arrangements with Generic Hypersurfaces Added
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917881283870720 |
|---|---|
| author | Reinke, Bernhard Wang, Kexin |
| author_facet | Reinke, Bernhard Wang, Kexin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20869 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hypersurface Arrangements with Generic Hypersurfaces Added Reinke, Bernhard Wang, Kexin Algebraic Geometry 05E14, 14C17, 52C35, 90C05, 90C51 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. |
| title | Hypersurface Arrangements with Generic Hypersurfaces Added |
| topic | Algebraic Geometry 05E14, 14C17, 52C35, 90C05, 90C51 |
| url | https://arxiv.org/abs/2412.20869 |