半代數集上的光滑點（ CS SC ）
- 2020 年 3 月 27 日
- 筆記
許多確定實代數或半代數集性質的算法依賴於計算光滑點的能力。現有的計算半代數集上光滑點的方法都使用符號量詞消除工具。摘要本文提出了一種簡單的算法，該算法通過計算某些精心選擇的函數的臨界點來保證實(半)代數集上每個連通緊分量上光滑點的計算。我們的技術在原理上是直觀的，在先前的複雜問題中表現良好，並且可以直接使用現有的數值代數幾何軟件實現。通過求解[數學處理誤差]情況下的庫拉莫托模型的平衡態數，證明了該方法的實際有效性。我們還設計了一種有效的算法來計算(半)代數集的實維數，而這也是本研究的初衷。
原文題目：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
