论文标题
半代数集的平滑点
Smooth Points on Semi-algebraic Sets
论文作者
论文摘要
许多用于确定实际代数或半代数集的属性的算法都取决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化器消除工具。在本文中，我们基于计算某些精心挑选函数的临界点提供了一种简单的算法，该函数可以确保计算真实（半）代数集的每个连接的紧凑组件中的平滑点。我们的技术在本金方面是直观的，在以前的困难示例上表现良好，并且可以直接使用现有的数值代数几何软件实现。通过解决$ n = 4 $案例的库拉莫托模型平衡数量的猜想来证明我们方法的实际效率。我们还应用我们的方法设计有效的算法来计算（半）代理集的实际维度，这是这项研究的原始动机。
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.
