Algebraic and complex geometry · Result 059

Counterexamples to Zariski’s multiplicity conjecture

2 papers

Disproves Zariski's multiplicity conjecture by constructing reduced holomorphic hypersurface germs that are ambiently homeomorphic but have different multiplicities. The examples include hypersurfaces in ℂ⁴ with isolated critical points and multiplicities four and five.

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▶Introduction: the problem and why it mattersin preparation

▶The presented solutionin preparation

Papers 2

Ambiently homeomorphic isolated hypersurfaces of multiplicities two and three

OpenAI · September 24, 2026

We give a negative answer to the embedded Zariski multiplicity conjecture. We construct two reduced hypersurface germs that are ambiently homeomorphic but have multiplicities two and three. Both have isolated singularities and lie in a common complex affine space of dimension divisible by eight. Their defining function germs are also topologically right equivalent, and the same examples answer Arnold's corank problem negatively for ambient topological equivalence.

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@misc{OAI:Ambiently-homeomorphic-isolated-hypersurfaces-of-multiplicities-two-and-three-September-24-2026,
  author = {{OpenAI}},
  title = {{Ambiently homeomorphic isolated hypersurfaces of multiplicities two and three}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Ambiently-homeomorphic-isolated-hypersurfaces-of-multiplicities-two-and-three-September-24-2026/paper.pdf}{OAI:Ambiently-homeomorphic-isolated-hypersurfaces-of-multiplicities-two-and-three-September-24-2026}},
  year = {2026}
}

Ambiently homeomorphic isolated hypersurface germs in ℂ⁴ with multiplicities four and five

OpenAI · September 27, 2026

We give a negative answer to Zariski's multiplicity question in four complex variables. We construct two reduced convergent holomorphic function germs with isolated critical points and multiplicities four and five whose zero-set germs are ambiently homeomorphic.

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Cite

@misc{OAI:Ambiently-homeomorphic-isolated-hypersurface-germs-in-C4-with-multiplicities-four-and-five-September-27-2026,
  author = {{OpenAI}},
  title = {{Ambiently homeomorphic isolated hypersurface germs in $\mathbb{C}^4$ with multiplicities four and five}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Ambiently-homeomorphic-isolated-hypersurface-germs-in-C4-with-multiplicities-four-and-five-September-27-2026/paper.pdf}{OAI:Ambiently-homeomorphic-isolated-hypersurface-germs-in-C4-with-multiplicities-four-and-five-September-27-2026}},
  year = {2026}
}

Formal verification

No Lean formalization is listed for this result yet. The repository notes that unformalized results could have issues.