Beyond Harnack Rigidity — Draconitic Research

Abstract

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 𝒞(𝒜_f)=∂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.

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BibTeX RIS

BibTeX

@misc{nisse2026beyond,
  title = {{Beyond Harnack Rigidity}},
  author = {Mounir Nisse},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.29655},
  eprint = {2609.29655},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.29655},
}

RIS

TY  - UNPB
TI  - Beyond Harnack Rigidity
AU  - Mounir Nisse
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.29655
UR  - https://arxiv.org/abs/2609.29655
AB  - 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 𝒞(𝒜_f)=∂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.
ER  -

Abstract

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