Browse › Algebraic and complex geometry › 059 Counterexamples to Zariski’s multiplicity conjecture
Ambiently homeomorphic isolated hypersurfaces of multiplicities two and three
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Abstract
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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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- GitHub user acprk, R11: Zariski multiplicity counterexample, ambient dimension (GitHub repository openai-math-remarks, undated; literature search cutoff 2026-10-09). Feeds a shorter integer product relation, found by lattice reduction, into this paper's construction and reports ambient dimension N = 1,802,976 instead of about 4.6·10^143; it states that only the finite arithmetic was checked and that the topology and lattice theory were not re-proved. [AI-generated per source]
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]
- Alexandre Fernandes, Zbigniew Jelonek and José Edson Sampaio, Proof of the metric Arnold’s corank problem (arXiv:2509.03447v2, 17 December 2025). The AI Proved What? exposition cites its invariance of multiplicity two under ambient bi-Lipschitz equivalence to explain the distinction from this manuscript’s claimed topological equivalence.
Possibly relevant literature (found by the agent, not checked)
- 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 equimultiplicity theorem for μ-constant families, which the paper uses to conclude that no μ-constant family connects its two germs.
- Lev Birbrair, Alexandre Fernandes, J. Edson Sampaio, Misha Verbitsky, Multiplicity of singularities is not a bi-Lipschitz invariant (arXiv:1801.06849; Math. Ann. 377 (2020)). Constructs bi-Lipschitz equivalent complex algebraic singularities with different multiplicities; relevant to the paper's comparison of its topological counterexample with metric results (Fernandes–Jelonek–Sampaio).
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
- VibeMathed: Problem record
- VibeMathed: Problem record
- VibeMathed: Problem record
- Emergent Mind: family 059
Independent sites; their content is not reviewed here.
Cite
@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}
}From OpenAI's README for this paper:
Run the verification code
Both checks use only the Python standard library. From this paper directory, run:
env -u PYTHONOPTIMIZE python3 verification/support/supplied_finite_rank.py
verification_tmp=$(mktemp -d)
python3 verification/validation/explicit_relation.py verification/support/explicit-relation.json "$verification_tmp/explicit-relation-verification.json"
Run the first command with assertions enabled, without Python's -O option; it exits silently when the cutoff counts and rank certificate pass. The second command verifies the explicit integer relation and the construction's variable counts, and writes a JSON report. These checks cover the finite arithmetic, not the lattice-theoretic or topological arguments.
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: