发表机构
Xiamen University Malaysia（厦门大学马来西亚分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明平面变形虫边界与轮廓相等并非Harnack曲线的充分条件，构造了反例并给出非环面等价族，否定了轮廓-边界刚性，细化了补偿问题。
AI 中文摘要
平面变形虫的边界总是包含在其轮廓中，而相等是简单Harnack曲线的特征。我们证明，即使在强光滑性和非退化假设下，其逆命题也不成立。对于每个二维格点多边形，我们构造一条光滑的Newton非退化曲线，其对数临界轨迹光滑且嵌入轮廓满足$\mathcal C(\mathscr A_f)=\partial\mathscr A_f$，尽管该曲线不是Harnack曲线。我们还提供了明确的原始族和非原始族，它们与简单Harnack曲线不是环面等价的。一个具体的原始例子通过精确消元和Sturm根计数得到验证。这些结果否定了轮廓-边界刚性，并表明轮廓作为集合不能检测实结构或重合临界片的多重性，从而细化了Lang、Shapiro和Shustin提出的补偿问题。
英文摘要
The boundary of a plane amoeba is always contained in its contour, and equality is a characteristic feature of simple Harnack curves. We show that the converse fails, even under strong smoothness and nondegeneracy assumptions. For every two-dimensional lattice polygon, except unimodular triangles, we construct a smooth Newton-nondegenerate curve with smooth logarithmic critical locus and smooth embedded contour satisfying $\mathcal C(\mathscr A_f)=\partial\mathscr A_f$, although the curve is not Harnack. We also provide explicit primitive and nonprimitive families that are not torus-equivalent to simple Harnack curves. A concrete primitive example is certified by exact elimination and Sturm root counting. These results disprove contour--boundary rigidity and show that the contour as a set does not detect the real structure or the multiplicity of coincident critical sheets, thereby refining the compensation problem proposed by Lang, Shapiro, and Shustin.
Comments36 page, 7 figures,; all comments are welcome
