Let $\mathbb{R}$ be the field of real numbers. We consider the problem of computing the real isolated points of a real algebraic set in $\mathbb{R}^n$ given as the vanishing set of a polynomial system. This problem plays an important role for studying rigidity properties of mechanism in material designs. In this paper, we design an algorithm which solves this problem. It is based on the computations of critical points as well as roadmaps for answering connectivity queries in real algebraic sets. This leads to a probabilistic algorithm of complexity $(nd)^{O(n\log(n))}$ for computing the real isolated points of real algebraic hypersurfaces of degree $d$. It allows us to solve in practice instances which are out of reach of the state-of-the-art.

中文翻译：

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计算代数超曲面的实孤立点

令 $\mathbb{R}$ 为实数域。我们考虑计算 $\mathbb{R}^n$ 中实代数集的实孤立点的问题，该实代数集作为多项式系统的消失集。该问题对于研究材料设计中机构的刚性特性具有重要作用。在本文中，我们设计了一种算法来解决这个问题。它基于关键点的计算以及用于回答真实代数集中的连通性查询的路线图。这导致了复杂度为 $(nd)^{O(n\log(n))}$ 的概率算法，用于计算 $d$ 次实代数超曲面的实孤立点。它使我们能够在实践中解决最先进技术无法实现的实例。