论文标题
复杂的链接和希尔伯特 - 塞缪尔的多重性
Complex Links and Hilbert-Samuel Multiplicities
论文作者
论文摘要
我们描述了一个框架，用于估算Hilbert-Samuel Mulutmities $ e_xy $，用于从有限点样本而不是定义方程式中的一对投影品品种$ x \ subset y $的框架。第一步涉及证明，在某些超平面部分下，这种多重性仍然不变，该部分将$ x $降低到$ p $ $ p $，而$ y $ to curve $ c $。接下来，我们确定$ e_pc $等于$ p $ $ c $的复杂链接的欧拉特征（因此，是基数）。最后，我们对所需的统一点样本数量（在$ c $中的$ p $）中提供明确的界限，以高信任地确定这种欧拉的特征。
We describe a framework for estimating Hilbert-Samuel multiplicities $e_XY$ for pairs of projective varieties $X \subset Y$ from finite point samples rather than defining equations. The first step involves proving that this multiplicity remains invariant under certain hyperplane sections which reduce $X$ to a point $p$ and $Y$ to a curve $C$. Next, we establish that $e_pC$ equals the Euler characteristic (and hence, the cardinality) of the complex link of $p$ in $C$. Finally, we provide explicit bounds on the number of uniform point samples needed (in an annular neighborhood of $p$ in $C$) to determine this Euler characteristic with high confidence.
