Browse › Algebraic and complex geometry › 059 Counterexamples to Zariski’s multiplicity conjecture
Ambiently homeomorphic isolated hypersurface germs in ℂ⁴ with multiplicities four and five
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Abstract
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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Related prior literature mentioned by others
- Mark McLean, Floer Cohomology, Multiplicity and the Log Canonical Threshold (Geometry & Topology 23, 2019). The AI Proved What? exposition invokes this contact-link invariance theorem to distinguish the stronger contact equivalence from the ambient topological equivalence claimed by both manuscripts. [family-level]
Possibly relevant literature (found by the agent, not checked)
- José Edson Sampaio, On the Arnold corank, the Zariski multiplicity, and the tangent Milnor fiber problems (arXiv:2610.19357, 14 Oct 2026; first posted on ResearchGate in August 2026). Proves corank-mod-2 invariance and multiplicity invariance for right topological equivalence of generalized semi-Brieskorn–Pham germs, including a multiplicity-two statement in ℂ³; relevant to the paper's remark that the three-variable ambient case remains untreated.
- Shamuel Auyeung, Adjacent Singularities, TQFTs, and Zariski's Multiplicity Conjecture (arXiv:2308.13925, 2023). Gives a Floer-theoretic alternative proof of the Fernández de Bobadilla–Pełka family theorem, which the paper invokes to conclude that its germs cannot be endpoints of a μ-constant path.
Digests and expositions
- AI Proved What?, Same knot, other degree? Counterexamples to Zariski’s multiplicity conjecture (Episode 059, October 2026). This video and accompanying transcript outline both proposed constructions through integral Seifert forms, strong approximation and link classification, and distinguish their dimensions and multiplicities. [AI-generated per source] [family-level]
Elsewhere
- Emergent Mind: family 059
Independent sites; their content is not reviewed here.
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}
}Family 059: Counterexamples to Zariski’s multiplicity conjecture
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.
Other papers in this family:
- Ambiently homeomorphic isolated hypersurfaces of multiplicities two and three · September 24, 2026