Smooth points on semi-algebraic sets
Smooth points on semi-algebraic sets
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DOI：
10.1016/j.jsc.2022.09.003
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发表时间：
2020-02
影响因子：
0.1
通讯作者：
Katherine Harris;J. Hauenstein;Á. Szántó
中科院分区：
文献类型：
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作者：
Katherine Harris;J. Hauenstein;Á. Szántó
许多确定半代数集性质的算法依赖于计算光滑点的能力[1]。我们提出了一个简单的程序的基础上计算的临界点的一些精心挑选的功能，保证光滑点的计算在每个连接有界组件的真实的原子半代数集。我们的技术原则上是直观的，在以前困难的例子中表现良好，并且使用现有的数值代数几何软件可以直接实现。证明了我们的方法的实际效率，通过解决一个猜想的平衡点的数量的仓本模型，然后= 4的情况下。我们也应用我们的方法来设计一个有效的算法来计算代数集的真实的维数，这是本研究的原始动机。
Many algorithms for determining properties of semi-algebraic sets rely upon the ability to compute smooth points [1]. We present a simple procedure based on computing the critical points of some well-chosen function that guarantees the computation of smooth points in each connected bounded component of a real atomic 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 then= 4 case. We also apply our method to design an efficient algorithm to compute the real dimension of algebraic sets, the original motivation for this research.