Segre-driven radicality testing

Abstract

We present a probabilistic algorithm to test if a homogeneous polynomial ideal I defining a scheme X in $P^n$ 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 components outside of the singular locus of $X_{\mathrm{red}}=V\left(\sqrt{I}\right)$, 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.