Segre-driven radicality testing

We present a probabilistic algorithm to test if a homogeneous polynomial ideal I defining a scheme X in Pn is radical using Segre classes and other geometric notions from intersection theory which is applicable for certain classes of ideals. If all isolated primary components of the scheme X are reduced and it has no embedded root components outside of the singular locus of Xred = V( I), then the algorithm is not applicable and will return that it is unable to decide radically; in all the other cases it will terminate successfully and in either case its complexity is singly exponential in n. The realm of the ideals for which our radical testing procedure is applicable and for which it 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. (c) 2023 Elsevier Ltd. All rights reserved.

Automated summary

We propose a probabilistic algorithm to test if a homogeneous polynomial ideal I defining a scheme X in Pn is radical. This algorithm utilizes Segre classes and other geometric notions from intersection theory and is applicable for certain classes of ideals. The algorithm terminates successfully with singly exponential complexity in n except in cases where all isolated primary components of X are reduced and there are no embedded root components outside of the singular locus of Xred = V(I), in which case it is unable to decide radically.