论文标题
代数流形的计算几何特征大小
Computing geometric feature sizes for algebraic manifolds
论文作者
论文摘要
我们介绍了数值代数几何方法，用于计算覆盖范围上的下限，局部特征大小以及使用该品种定义的多项式作为输入的等分和光滑代数变体的实际部分的弱特征大小。对于弱特征大小，我们还表明，非二次完成的交叉点通常具有有限的几何瓶颈，并描述了如何直接计算弱特征大小而不是在这种情况下的下限。在所有其他情况下，我们描述了可用于确定特征大小值而不是下限的其他计算。我们还提出了同源推理实验，将持续的同源性计算与我们功能尺寸算法的实现版本相结合，既有全球密集的样本和相对于局部特征大小的可适应性密度的样品。
We introduce numerical algebraic geometry methods for computing lower bounds on the reach, local feature size, and the weak feature size of the real part of an equidimensional and smooth algebraic variety using the variety's defining polynomials as input. For the weak feature size, we also show that non-quadratic complete intersections generically have finitely many geometric bottlenecks, and describe how to compute the weak feature size directly rather than a lower bound in this case. In all other cases, we describe additional computations that can be used to determine feature size values rather than lower bounds. We also present homology inference experiments that combine persistent homology computations with implemented versions of our feature size algorithms, both with globally dense samples and samples that are adaptively dense with respect to the local feature size.
