Faster computation of Whitney stratifications and their minimization

TL;DR

This paper introduces two new algorithms for efficiently computing Whitney stratifications of algebraic varieties, including the first algorithm for obtaining a minimal Whitney stratification, based on equidimensional decomposition and classical theoretical results.

Contribution

The paper presents novel algorithms that improve efficiency in computing Whitney stratifications and introduces the first method for minimizing a given stratification.

Findings

The modified algorithm is more efficient than previous methods.

The second algorithm can produce the minimal Whitney stratification.

This work is the first to algorithmically compute a minimal Whitney stratification.

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.