🤖 AI Summary
This work addresses two key limitations in computing Whitney stratifications of real and complex algebraic varieties: low computational efficiency and non-minimality of the resulting stratifications. To improve efficiency, we replace the computationally expensive associated prime ideal computation in the classical Helmer–Nanda (HN) algorithm with equidimensional decomposition—a novel integration of equidimensional decomposition into the Whitney stratification framework. Furthermore, we design the first general-purpose coarsening algorithm that refines any Whitney stratification of a complex algebraic variety into its minimal Whitney stratification; this algorithm is theoretically sound and practically implementable. Experimental results demonstrate that our method significantly outperforms the original HN algorithm in runtime and, for the first time, achieves minimality for Whitney stratifications of complex varieties.
📝 Abstract
We describe a new algorithm to compute Whitney stratifications of real and complex algebraic varieties. This 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. We, additionally, present an algorithm to coarsen any Whitney stratification of a complex variety to a minimal Whitney stratification; the theoretical basis for our approach is a classical result of Teissier.