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.

翻译：许多用于确定实代数集或半代数集性质的算法依赖于计算光滑点的能力。现有计算半代数集光滑点的方法使用符号量词消去工具。本文提出一种基于计算精心选择函数的临界点的简单算法，该算法保证在实（半）代数集的每个连通紧致分支上都能计算出光滑点。我们的方法在原则上直观易懂，在先前困难的示例上表现良好，且易于利用现有数值代数几何软件实现。通过解决$n=4$情况下Kuramoto模型平衡点数量的猜想，证明了该方法在实际中的高效性。我们还应用该方法设计了计算（半）代数集实维数的高效算法——这也是本研究的原始动机。