半代数集上的光滑点（ CS SC ）
- 2020 年 3 月 27 日
- 筆記
许多确定实代数或半代数集性质的算法依赖于计算光滑点的能力。现有的计算半代数集上光滑点的方法都使用符号量词消除工具。摘要本文提出了一种简单的算法，该算法通过计算某些精心选择的函数的临界点来保证实(半)代数集上每个连通紧分量上光滑点的计算。我们的技术在原理上是直观的，在先前的复杂问题中表现良好，并且可以直接使用现有的数值代数几何软件实现。通过求解[数学处理误差]情况下的库拉莫托模型的平衡态数，证明了该方法的实际有效性。我们还设计了一种有效的算法来计算(半)代数集的实维数，而这也是本研究的初衷。
原文题目：Smooth Points on Semi-algebraic Sets
原文：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 [Math Processing Error] 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.
原文作者：Katherine Harris, Jonathan D. Hauenstein, Agnes Szanto
原文地址：http://cn.arxiv.org/abs/2002.04707
