Isolated-singularity version of Zariski's multiplicity conjecture

About 1 year old · traced to 2025

Let $f,g\colon(\mathbb{C}^n,0)\to(\mathbb{C},0)$ be holomorphic function-germs with isolated singularities at the origin. Suppose there is a homeomorphism $\varphi\colon(\mathbb{C}^n,V(f),0)\to(\mathbb{C}^n,V(g),0)$ . Isolated-singularity multiplicity conjecture. Then

m(V(f),0)=m(V(g),0).

The paper states that the general Zariski multiplicity problem remains open even in this isolated-singularity setting; only a result for families with isolated singularities is reported as confirmed.

References

Primary source

José Edson Sampaio, “On the Milnor fibres of initial forms of topologically equivalent holomorphic functions”, arXiv:2503.17127 (2025).

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RemarkAI-assistedClaimed by OpenAI. The manuscript claims two reduced weighted-homogeneous hypersurface germs with isolated critical points in a finite complex ambient dimension greater than three and divisible by eight. Their ambient zero-set pairs are homeomorphic, but their multiplicities are two and three. This gives a claimed counterexample to the isolated-singularity multiplicity statement; no equisingular family is asserted.See full solutionHide full solution

Claimed by OpenAI. The manuscript claims two reduced weighted-homogeneous hypersurface germs with isolated critical points in a finite complex ambient dimension greater than three and divisible by eight. Their ambient zero-set pairs are homeomorphic, but their multiplicities are two and three. This gives a claimed counterexample to the isolated-singularity multiplicity statement; no equisingular family is asserted.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Ambiently-homeomorphic-isolated-hypersurfaces-of-multiplicities-two-and-three-September-24-2026/paper.pdf

OpenAI-059-01-Ambiently-homeomorphic-isolated-hypersurfaces-of-multiplicities-two-and-three.pdf560,543 bytesOpen

RemarkAI-assistedClaimed by OpenAI. The manuscript claims two reduced convergent hypersurface germs in complex four-space with isolated critical points and multiplicities four and five, whose ambient zero-set pairs are homeomorphic. This gives a claimed counterexample to the isolated-singularity multiplicity statement. The pair is not asserted to be connected by a constant-Milnor-number family.See full solutionHide full solution

Claimed by OpenAI. The manuscript claims two reduced convergent hypersurface germs in complex four-space with isolated critical points and multiplicities four and five, whose ambient zero-set pairs are homeomorphic. This gives a claimed counterexample to the isolated-singularity multiplicity statement. The pair is not asserted to be connected by a constant-Milnor-number family.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Ambiently-homeomorphic-isolated-hypersurface-germs-in-C4-with-multiplicities-four-and-five-September-27-2026/paper.pdf

OpenAI-059-02-Ambiently-homeomorphic-isolated-hypersurface-germs-in-with-multiplicities-four-and-five.pdf410,248 bytesOpen