Zariski's multiplicity conjecture
From OpenAI Aftermath
Zariski's multiplicity conjecture asks whether the multiplicity of a complex hypersurface germ, the lowest degree of a term of its defining equation, is determined by the embedded topology of the germ: whether two reduced hypersurface germs related by a homeomorphism of ambient neighborhoods must have the same multiplicity. Zariski posed the question in 1971. Two manuscripts dated 24 and 27 September 2026 in OpenAI's public mathematics collection (github.com/openai/math) claim counterexamples: ambiently homeomorphic germs with isolated singularities and multiplicities two and three, in a space of dimension divisible by eight, and multiplicities four and five in four variables. The collection has no Lean formalization of these results. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.
Background
- As recounted in the introductions of the manuscripts:
- For plane curves the embedded topology determines the characteristic data of the branches and their intersection numbers, and hence the multiplicity (Zariski).
- Within deformations, Greuel and O'Shea proved equimultiplicity for Milnor-number-constant deformations of isolated quasihomogeneous singularities, and Fernández de Bobadilla and Pełka proved that every continuous family of power series with constant finite Milnor number has constant multiplicity; the claimed counterexamples therefore cannot be joined by such a family.
- For pairs under extra hypotheses, Yau proved invariance for isolated quasihomogeneous surface singularities in C3, and Fernandes, Jelonek and Sampaio proved that multiplicity two is preserved by ambient bi-Lipschitz equivalence and that the Hessian corank is preserved by ambient homeomorphisms in three variables. Sampaio's corank-parity theorem gives an ambient topological invariant in every dimension.
- Arnold's corank problem asks whether the corank of the Hessian of a function germ is a topological invariant.
The original results
- Claims.
- Multiplicities two and three. There are two reduced hypersurface germs with isolated singularities, in a complex affine space of dimension divisible by eight, that are ambiently homeomorphic but have multiplicities two and three. Their defining germs are also topologically right equivalent and have different Hessian coranks, which the manuscript presents as a negative answer to Arnold's corank problem for ambient topological equivalence; it states that the two coranks have the same parity, as Sampaio's theorem requires.
- Four variables. There are reduced convergent function germs f4, f5 on C4 with isolated critical points, of multiplicities four and five, whose zero sets are ambiently homeomorphic.
- Method, as described: the germs are compared through their Milnor open books. For the open books used, a classification theorem turns a congruence of integral Seifert forms into an equivalence of the embedded links, so the work is to obtain that integral congruence while keeping different orders of the equations. The first manuscript uses Thom–Sebastiani sums of chain-type and power singularities and ships an explicit integral relation (849 terms) with a validator.
- Sources (openai/math repository, commit
1581d4c):- Ambiently homeomorphic isolated hypersurfaces of multiplicities two and three (24 September 2026);
- Ambiently homeomorphic isolated hypersurface germs in C4 with multiplicities four and five (27 September 2026).
- Formalization. The collection has no Lean scope note or Comparator challenge for this family (checked 21 October 2026).
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 21 October 2026) |
|---|---|---|---|---|
| 2026-10-09 | Pico (Håkon Åmdal), row 059 in piiiico/openai-math-counterexamples (GitHub; commit 01966fb) |
Verification (certificate) | Checks the explicit relation shipped with the multiplicities-two-and-three manuscript (849 terms) by two routes independent of the authors' validator, a prime-valuation sieve and reduction modulo random primes: the weighted sum is positive and both product relations hold. Reports "holds"; altered relations are rejected. | Part of a census of counterexample families in the collection. It checks the finite certificate, not the topological classification steps. |
| 2026-10-13 (arXiv version 1) | J. E. Sampaio, On the Arnold corank, the Zariski multiplicity, and the tangent Milnor fiber problems (arXiv:2610.19357) | Related (an earlier version is cited by the manuscripts) | Proves that the Hessian corank is invariant modulo 2 under ambient homeomorphisms of reduced hypersurface germs, and that multiplicity, when different from two, is invariant for generalized semi-Brieskorn–Pham hypersurfaces. It shows that a weak form of a conjecture of Budur, Fernández de Bobadilla, Lê and Nguyen on Milnor fibers of initial forms is equivalent to Zariski's conjecture for right topological equivalence. The multiplicities-two-and-three manuscript cites a 28 August version and states that its examples are consistent with the parity theorem. | arXiv preprint, version 1. It treats Zariski's question as open and does not mention the collection. |
| 2026-10-16 | J. Jardine, Lattice24, referee reviews rank_376 (multiplicities two and three) and rank_255 (four variables) in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995) | Review (AI-assisted audit) | Both reviews report no gap; the reviewers state that they read every section.
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Part of an audit of the collection's manuscripts that the audit describes as complete. The reviews are published under CC BY 4.0. |
Notes
- The earlier results are due to the authors named in the Background section, as recounted by the manuscripts.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
