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.

翻译：我们提出了一个概率算法来测试,如果一个单一的多元多边理想用美元来界定一个以美元计的公式($mathbb{P ⁇ n$),使用Segre等级和其他交叉理论的几何概念是激进的,其最坏的情况复杂性取决于X$的几何。如果这个方案用美元减少了孤立的初级成分,而没有嵌入组件支持美元为X ⁇ rm red ⁇ V(\sqrt{I})$的单极点,那么最坏的情况复杂性以美元计倍增倍;在其他所有情况中,复杂性是单倍指数化的。我们激进测试程序只需要单倍指数时间的理想领域包括常常被视为病理性的例子,例如从著名的Mayr-Meyer系列理想中提取的模型,这些模型显示理想会籍问题具有双倍指数复杂性。