AI 中文总结
该研究提出基于二阶必要条件的算法测试代数集上点的孤立性，进而界定代数集在某点的局部维数，其计算避免经典方法的组合增长，经应用完成了正则4维多面体局部维数计划。
AI 中文摘要
我们提出一种基于二阶必要条件的算法，用于测试代数集上的一个点是否为孤立点。利用该子程序，我们开发出一种算法，可界定代数集在某点处的局部维数。我们的孤立测试中的主要计算由雅可比矩阵的零空间决定，避免了经典方法的组合式增长。对于精确输入，该算法返回的界是可验证的。我们将此方法与经典方法进行了比较。作为应用，我们证明了24胞体的实现空间在其对称正则实现处具有预期的局部维数，完成了Rastanawi、Sinn和Ziegler发起的正则4维多面体局部维数计划。
英文摘要
We give an algorithm, based on second-order necessary conditions, to test whether a point is isolated on an algebraic set. Using this subroutine, we develop an algorithm that bounds the local dimension of an algebraic set at a point. The principal computation in our isolation test is governed by the nullity of the Jacobian, avoiding the combinatorial growth of classical methods. For exact input, the bound returned by the algorithm is certified. We compare our approach with classical methods. As an application, we show that the realization space of the 24-cell has the expected local dimension at its symmetric regular realization, completing the local dimension program for regular 4-polytopes initiated by Rastanawi, Sinn, and Ziegler.
Comments17 pages, 1 figure
