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Faster computation of Whitney stratifications and their minimization

Martin Helmer, Rafael Mohr

TL;DR

The paper addresses the computational bottleneck in constructing Whitney stratifications by introducing a fast algebraic criterion that avoids full primary decomposition in conormal spaces, and a complementary minimization approach based on Teissier's local polar varieties to obtain the unique minimal Whitney stratification for complex varieties. The first algorithm accelerates stratification by using equidimensional decomposition, while the second leverages polar multiplicities to ensure minimality. The authors provide a probabilistic, multiplicity-based minimization pipeline and prove correctness, along with runtime benchmarks showing notable speedups over previous methods. A Macaulay2 implementation demonstrates practical applicability to nontrivial examples, including those challenging for prior algorithms.

Abstract

We describe two new algorithms for the computation of Whitney stratifications of real and complex algebraic varieties. The first algorithm is a modification of the algorithm of Helmer and Nanda (HN), but is made more efficient by using techniques for equidimensional decomposition rather than computing the set of associated primes of a polynomial ideal at a key step in the HN algorithm. We note that this modified algorithm may fail to produce a minimal Whitney stratification even when the HN algorithm would produce a minimal stratification. The second algorithm coarsens a given Whitney stratification of a complex variety to the unique minimal Whitney stratification; we refer to this as the minimization of a stratification. The theoretical basis for our approach is a classical result of Teissier. To our knowledge this yields the first algorithm for computing a minimal Whitney stratification.

Faster computation of Whitney stratifications and their minimization

TL;DR

The paper addresses the computational bottleneck in constructing Whitney stratifications by introducing a fast algebraic criterion that avoids full primary decomposition in conormal spaces, and a complementary minimization approach based on Teissier's local polar varieties to obtain the unique minimal Whitney stratification for complex varieties. The first algorithm accelerates stratification by using equidimensional decomposition, while the second leverages polar multiplicities to ensure minimality. The authors provide a probabilistic, multiplicity-based minimization pipeline and prove correctness, along with runtime benchmarks showing notable speedups over previous methods. A Macaulay2 implementation demonstrates practical applicability to nontrivial examples, including those challenging for prior algorithms.

Abstract

We describe two new algorithms for the computation of Whitney stratifications of real and complex algebraic varieties. The first algorithm is a modification of the algorithm of Helmer and Nanda (HN), but is made more efficient by using techniques for equidimensional decomposition rather than computing the set of associated primes of a polynomial ideal at a key step in the HN algorithm. We note that this modified algorithm may fail to produce a minimal Whitney stratification even when the HN algorithm would produce a minimal stratification. The second algorithm coarsens a given Whitney stratification of a complex variety to the unique minimal Whitney stratification; we refer to this as the minimization of a stratification. The theoretical basis for our approach is a classical result of Teissier. To our knowledge this yields the first algorithm for computing a minimal Whitney stratification.

Paper Structure

This paper contains 10 sections, 14 theorems, 34 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Gröbner Bases
  4. Whitney Stratifications
  5. Local Polar Varieties and Minimal Whitney Stratifications
  6. New Whitney Stratification Algorithms
  7. A New Algebraic Criterion for Checking Whitney's Condition
  8. The Whitney Stratification Algorithm
  9. The Minimization Algorithm
  10. Runtime Tests

Key Result

Proposition 2.5

Let $\mathbf{x}$ and $\mathbf{y}$ be any two finite sets of variables and let $\prec$ be any block order eliminating $\mathbf{x}$. Let $G$ be a Gröbner basis of some ideal $I$ with respect to $\prec$. Then

Figures (2)

  • Figure 1: Plots of the curve defined by \ref{['eq:planarCubic']} for different parameter values $z$; the topology of the curve changes at $z=1$. While the curve in (\ref{['fig:smooth']}) is smooth and has two connected components (of different dimensions) in $\mathbb{R}^2$ it is connected and singular, with singularity at $(1,1)$, in $\mathbb{C}^2$.
  • Figure 2: The surface $X=\mathbf{V}\left((y-1)^2-(x-z)(x-1)^2 \right)$ in $\mathbb{R}^3$. It's singular locus is $Y=\mathbf{V}(x-1, y-1)$ and its Whitney stratification arises from the flag $\{(1,1,1)\} \subset Y \subset X$. The dimension 2 stratum is $X-Y$, the two dimension 1 strata are the connected components of $Y-\{(1,1,1)\}$, and the dimension 0 stratum is $(1,1,1)$.

Theorems & Definitions (42)

  • Example 1.1
  • Definition 2.1: Monomial Order
  • Definition 2.2: Leading Monomial
  • Definition 2.3: Gr√∂bner Basis
  • Definition 2.4: Block Order
  • Proposition 2.5
  • Definition 2.6: Maximally Independent Subset
  • Proposition 2.7
  • Definition 2.8: Whitney's Condition (B)
  • Definition 2.9: Whitney Stratification
  • ...and 32 more