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Computing Arrangements of Hypersurfaces

Paul Breiding, Bernd Sturmfels, Kexin Wang

TL;DR

The paper introduces HypersurfaceRegions.jl for computing all regions of the complement of an arrangement of real algebraic hypersurfaces in $\mathbb{R}^n$, with regions organized by sign patterns $\sigma\in\{-1,+1\}^k$ and their Euler characteristics. The core method builds an exhaustive Morse function $g$ from a master function $g(x)=\dfrac{\prod|f_i|^{s_i}}{q^t}$, locates critical points via $\nabla\log g=0$ using $HomotopyContinuation.jl$, and uses the Mountain Pass Theorem to connect index-0 points through index-1 points by gradient flow, thereby identifying region boundaries and invariants. The approach also includes a projective variant and a user-friendly Julia interface, enabling applications to generic hypersurface systems and spectrahedra with experiments indicating robust scalability and accurate region counts (including noncontractible regions). This provides a practical tool to study real hypersurface arrangements, compute Euler characteristics per region, and explore discriminantal and ML-degree related topology in higher dimensions. The work demonstrates how numerical algebraic geometry, Morse theory, and path tracking yield a versatile pipeline for topological classification of real algebraic complements.

Abstract

We present a Julia package HypersurfaceRegions.jl for computing all connected components in the complement of an arrangement of real algebraic hypersurfaces in $\mathbb{R}^n$.

Computing Arrangements of Hypersurfaces

TL;DR

The paper introduces HypersurfaceRegions.jl for computing all regions of the complement of an arrangement of real algebraic hypersurfaces in , with regions organized by sign patterns and their Euler characteristics. The core method builds an exhaustive Morse function from a master function , locates critical points via using , and uses the Mountain Pass Theorem to connect index-0 points through index-1 points by gradient flow, thereby identifying region boundaries and invariants. The approach also includes a projective variant and a user-friendly Julia interface, enabling applications to generic hypersurface systems and spectrahedra with experiments indicating robust scalability and accurate region counts (including noncontractible regions). This provides a practical tool to study real hypersurface arrangements, compute Euler characteristics per region, and explore discriminantal and ML-degree related topology in higher dimensions. The work demonstrates how numerical algebraic geometry, Morse theory, and path tracking yield a versatile pipeline for topological classification of real algebraic complements.

Abstract

We present a Julia package HypersurfaceRegions.jl for computing all connected components in the complement of an arrangement of real algebraic hypersurfaces in .
Paper Structure (5 sections, 6 theorems, 25 equations, 6 figures, 3 tables, 2 algorithms)

This paper contains 5 sections, 6 theorems, 25 equations, 6 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Log-likelihood as a Morse function
  3. Mountain Passes and Path Tracking
  4. How to Use the Software
  5. Experiments

Key Result

Theorem 3

The manifold $\mathcal{M}$ is homotopy equivalent to a CW-complex with exactly one $\ell$-cell for every critical point of index $\ell$. In particular, the Euler characteristic of $\mathcal{M}$ equals where $\mu_\ell$ is the number of index $\ell$ critical points of the exhaustive Morse function $g : \mathcal{M} \rightarrow \mathbb{R}$.

Figures (6)

  • Figure 1: A skinny ellipse pierces two concentric spheres. This arrangement has $8$ regions.
  • Figure 2: A hyperboloid and two paraboloids. The paraboloids intersect in a circle that spans a plane. This arrangement of four surfaces has $12$ regions in $\mathbb{R}^3$. Four are contractible.
  • Figure 3: The output of HypersurfaceRegions.jl for the arrangement shown in Figure \ref{['fig:hyperboloid']}.
  • Figure 4: Output for the discriminantal arrangement in Example \ref{['ex:dis8']}.
  • Figure 5: Arrangement given by the elliptope and the facet planes of the surrounding cube.
  • ...and 1 more figures

Theorems & Definitions (16)

  • Example 1: $n\!=\!k\!=\!3$
  • Example 2: $n=2,k=21$
  • Theorem 3
  • Proposition 4
  • Example 5
  • Proposition 6
  • Example 7
  • Corollary 8
  • Theorem 9
  • Corollary 10
  • ...and 6 more