Segre-driven radicality testing

Martin Helmer (1) , Elias Tsigaridas (2, 3, 4)

1 NCSU - Department of Mathematics [Raleigh]
2 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
3 OURAGAN - OUtils de Résolution Algébriques pour la Géométrie et ses ApplicatioNs
4 SU - Sorbonne Université

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. 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 Xred=V(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.

Domaines: Mathématiques [math]; Informatique [cs]

Martin Helmer, Elias Tsigaridas. Segre-driven radicality testing. Journal of Symbolic Computation, 2024, 122, pp.102262. ⟨10.1016/j.jsc.2023.102262⟩. ⟨hal-04222033⟩