Segre-Driven Radicality Testing

TL;DR

This paper introduces a probabilistic algorithm leveraging Segre classes to efficiently determine if a polynomial ideal is radical, with complexity depending on the geometric properties of the associated scheme.

Contribution

It presents a novel geometric approach to radicality testing using intersection theory, improving complexity bounds for certain classes of ideals including pathological cases.

Findings

Algorithm is probabilistic and geometric in nature.

Complexity varies from singly to doubly exponential based on scheme properties.

Includes examples like Mayr-Meyer ideals demonstrating efficiency.

Abstract

We present a probabilistic algorithm to test if a homogeneous polynomial ideal $I$ defining a scheme $X$ in $\mathbb{P}^n$ is radical using Segre classes and other geometric notions from intersection theory. Its worst case complexity depends on the geometry of $X$. If the scheme $X$ has reduced isolated primary components and no embedded components supported the singular locus of $X_{\rm red}=V(\sqrt{I})$, then the worst case complexity is doubly exponential in $n$; in all the other cases the complexity is singly exponential. The realm of the ideals for which our radical testing procedure requires only single exponential time includes examples which are often considered pathological, such as the ones drawn from the famous Mayr-Meyer set of ideals which exhibit doubly exponential complexity for the ideal membership problem.