Result 059, Algebraic and complex geometry

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.

Disproof or counterexample

The bigger picture

Why it matters

Can the shape of a complex geometric object determine how its equation begins? These manuscripts claim counterexamples: two objects can look identical under continuous changes of coordinates yet have different algebraic multiplicities.

What changes?

The manuscripts report pairs of reduced hypersurface germs: local zero sets of holomorphic functions without repeated components. A homeomorphism of the surrounding space identifies each pair, despite different multiplicities, the lowest nonzero total degrees of their defining equations. One pair lies in complex dimension four, with isolated critical points and multiplicities four and five. Another has isolated singularities and multiplicities two and three in a shared complex affine space whose dimension is divisible by eight.

What does that help mathematicians do?

If valid, these constructions rule out recovering multiplicity from the local topological shape of a hypersurface together with its surroundings, even for isolated singularities. This separates two ways researchers classify singular points: by continuous shape and by analytic equations. In particular, an ambient homeomorphism alone cannot justify transferring a multiplicity calculation from one hypersurface to another; additional information would be needed.

Are there practical applications?

The immediate value is foundational: researchers comparing topological and analytic descriptions of complex hypersurfaces gain explicit examples where those descriptions diverge. The four-variable example supplies a concrete setting for testing proposed classification principles. This is a limitation on what topological data can reveal, not a computational method or a demonstrated practical application.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Ambiently homeomorphic isolated hypersurfaces of multiplicities two and three

September 24, 2026 31 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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

September 27, 2026 32 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.