  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.
