论文标题
代数和热带奇异性超曲面的枚举
Enumeration of algebraic and tropical singular hypersurfaces
论文作者
论文摘要
我们开发了Mikhalkin的晶格路径算法的版本，用于用于任意程度和维度的投射性超曲面，该版本列举了通过适当的点的奇异热带突出表面。通过证明对应定理与晶格路径算法相结合，我们构建了一个$δ$尺寸线性$ d $ $ d $真实的超曲面，其中包含$ \ frac {1} {δ！} {Δ！}（γ_nd^n）通过递归公式。这与数字$ \ frac {1} {δ！} \ left（（（n+1）（d-1）（d-1）^n \ right）^δ+o \ left（d^{n（δ-Δ-1）}^$δ$δ$δ$Δ$Δ$Δ$Δ$Δ$δ$δ$δ$δ。在$δ= 1 $的情况下，我们给出了一个更好的主词。
We develop a version of Mikhalkin's lattice path algorithm for projective hypersurfaces of arbitrary degree and dimension, which enumerates singular tropical hypersurfaces passing through appropriate configuration of points. By proving a correspondence theorem combined with the lattice path algorithm, we construct a $δ$ dimensional linear space of degree $ d $ real hypersurfaces containing $\frac{1}{δ!} (γ_nd^n)^δ+ O(d^{nδ-1})$ hypersurfaces with $ δ$ real nodes, where $ γ_n $ are positive and given by a recursive formula. This is asymptotically comparable to the number $ \frac{1}{δ!} \left( (n+1)(d-1)^n \right)^δ+O\left(d^{n(δ-1)} \right) $ of complex hypersurfaces having $ δ$ nodes in a $ δ$ dimensional linear space. In the case $ δ=1 $ we give a slightly better leading term.
