Many algorithms for determining properties of real algebraic or semi-algebraic sets rely upon the ability to compute smooth points. Existing methods to compute smooth points on semi-algebraic sets use symbolic quantifier elimination tools. In this paper, we present a simple algorithm based on computing the critical points of some well-chosen function that guarantees the computation of smooth points in each connected compact component of a real (semi)-algebraic set. Our technique is intuitive in principal, performs well on previously difficult examples, and is straightforward to implement using existing numerical algebraic geometry software. The practical efficiency of our approach is demonstrated by solving a conjecture on the number of equilibria of the Kuramoto model for the $n=4$ case. We also apply our method to design an efficient algorithm to compute the real dimension of (semi)-algebraic sets, the original motivation for this research.

中文翻译：

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半代数集上的平滑点

许多用于确定实代数或半代数集属性的算法依赖于计算平滑点的能力。在半代数集上计算平滑点的现有方法使用符号量词消除工具。在本文中，我们提出了一种基于计算一些精心选择的函数的临界点的简单算法，该算法保证了实（半）代数集的每个连接紧凑组件中平滑点的计算。我们的技术原则上是直观的，在以前困难的例子上表现良好，并且使用现有的数值代数几何软件很容易实现。我们的方法的实际效率通过在 $n=4$ 的情况下解决 Kuramoto 模型的均衡数的猜想来证明。