Zariski's multiplicity conjecture: does embedded topology determine the multiplicity of a hypersurface singularity?

For a reduced convergent power series $f\in\mathbb{C}\{z_1,\ldots,z_N\}$ with $f(0)=0$ , the multiplicity of the hypersurface germ $V(f)$ is $\mathrm{ord}_0 f$ , the least degree of a monomial in $f$ . Two germs are ambiently homeomorphic if a homeomorphism of neighbourhoods of $0$ in $\mathbb{C}^N$ fixing $0$ carries one zero set onto the other. For plane curves the embedded topology determines multiplicity, and by A'Campo and Le a germ ambiently homeomorphic to a smooth one is smooth. Zariski asked in 1971 whether this holds in general: if $(\mathbb{C}^N,V(f),0)$ and $(\mathbb{C}^N,V(g),0)$ are homeomorphic, must $V(f)$ and $V(g)$ have the same multiplicity?

Result: Disproved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Construction
Field: Singularity theory; topology of hypersurface singularities
Posed by: Oscar Zariski, 'Some open questions in the theory of singularities', Bull. AMS 77 (1971)
Year posed: 1971
Years open: 55y
Solved: 2026-09-24
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Unreviewed
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 52 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Two independent counterexample constructions. September 24: for some finite $N>3$ divisible by eight, reduced weighted homogeneous real polynomials $f_1,f_2$ on $\mathbb{C}^N$ with isolated critical points, $\mathrm{ord}_0f_1=2$ , $\mathrm{ord}_0f_2=3$ , and $(\mathbb{C}^N,V(f_1),0)\cong(\mathbb{C}^N,V(f_2),0)$ ; the germs are also topologically right equivalent, so the right-equivalence version and its mod-2 version (Sampaio 2025, Conjectures 2 and Z mod 2) fail too. September 27: reduced convergent germs $f_4,f_5$ on $\mathbb{C}^4$ with isolated critical points, orders $4$ and $5$ , ambiently homeomorphic zero sets. Not addressed: hypersurfaces in $\mathbb{C}^3$ , and metric (bi-Lipschitz) or family versions, where positive results stand.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. Two manuscripts in the family, September 24 and September 27, 2026, which the September 27 paper describes as independent arguments. The September 24 paper ships Python scripts (standard library only) that check its finite integer arithmetic; the README says they do not cover the lattice-theoretic or topological arguments.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of each manuscript was read against Zariski's question as posed (reduced germs, ambient homeomorphism of pairs, multiplicity as order). The proofs were not refereed. No Lean formalization is listed in lean/formalization.yaml and lean/docs has no page for this family. The September 24 construction runs through Levine's classification of simple knots, strong approximation for special unitary groups and the Thom-Sebastiani and Sakamoto formulas; the ambient dimension N is not computed, only shown to exist and be divisible by eight. Its Python scripts check the finite integer arithmetic only, and were not run here. Both papers state that the surface case in C^3 is untreated.

Zariski's multiplicity conjecture: does embedded topology determine the multiplicity of a hypersurface singularity?

What was actually shown

What the AI did

Verification

Sources

Discussion