阿米巴轮廓的尖点与节点
Cusps and nodes of Amoeba Contours
中文总结 AI 辅助
该研究针对光滑平面曲线的阿米巴轮廓奇点，改进了Lang–Shapiro–Shustin的尖点界，提出了横向s-节点的多重性敏感界，结合多种代数几何方法揭示了尖点与多重节点的不同几何机制。
中文摘要 AI 辅助
我们为光滑平面曲线的阿米巴轮廓奇点建立了新的牛顿多边形界。我们的尖点估计改进了Lang–Shapiro–Shustin的四次界，在保留牛顿多边形的归一化面积、边界格点和方向宽度的同时，将其主导系数从8降至4。我们还获得了横向s-节点的新的多重性敏感界，带有自然衰减因子1/组合数(s,2)。证明结合了对数高斯映射、归一化纤维积、分歧理论和饱和对角外相交概型，揭示了控制尖点和多重节点的不同几何机制。
英文摘要
We establish new Newton-polygon bounds for the singularities of amoeba contours of smooth plane curves. Our cusp estimate refines the degree-four bound of Lang--Shapiro--Shustin, reducing its leading coefficient from $8$ to $4$ while retaining the normalized area, boundary lattice points, and directional widths of the Newton polygon. We also obtain a new multiplicity-sensitive bound for transverse $s$-nodes, with the natural decay factor $1/\binom{s}{2}$. The proofs combine logarithmic Gauss maps, normalized fiber products, ramification theory, and saturated off-diagonal intersection schemes, revealing the distinct geometric mechanisms governing cusps and multiple nodes.
