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Math-2239 · mathematics · Algebraic geometry

Zariski multiplicity conjecture on the topological equisingularity and equimultiplicity of varieties at singular points

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
If two complex hypersurface germs have the same embedded topological singularity type, must their multiplicities at the singular point agree? The multiplicity is the least degree of a nonzero term in a local defining equation. The related family version asks whether topological equisingularity forces equimultiplicity.

Formulation references: [S1], [S2].

Short English statement

If two complex hypersurface germs have the same embedded topological singularity type, must their multiplicities at the singular point agree? The multiplicity is the least degree of a nonzero term in a local defining equation. The related family version asks whether topological equisingularity forces equimultiplicity.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2239-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2239

Manuscript 1 (family 059, 2026-09-27): The primary main theorem gives reduced isolated complex hypersurface germs whose embedded topological pairs are homeomorphic but whose multiplicities differ. Isolated germs are allowed by both unrestricted problem-statement multiplicity statements, so one such pair disproves them.

Manuscript 2 (family 059, 2026-09-24): The primary main theorem gives reduced isolated complex hypersurface germs whose embedded topological pairs are homeomorphic but whose multiplicities differ. Isolated germs are allowed by both unrestricted problem-statement multiplicity statements, so one such pair disproves them.

Zariski multiplicity conjecture on the topological equisingularity and equimultiplicity of varieties at singular points

OpenAI's claim

Manuscript 1 (family 059, 2026-09-27): Two isolated hypersurface germs in four complex variables are claimed to have the same ambient topological type while their multiplicities are four and five. Manuscript 2 (family 059, 2026-09-24): Two isolated weighted-homogeneous hypersurface germs in a common finite dimension divisible by eight are claimed to be ambiently homeomorphic despite having multiplicities two and three.

Mathematical statement of the claimed result

Mathematical claim 1.
OpenAI claims reduced convergent holomorphic germs $f_4,f_5:(\mathbb C^4,0)\to(\mathbb C,0)$ for which zero is an isolated critical point in each case, such that \[ \operatorname{ord}_0f_4=4,\qquad\operatorname{ord}_0f_5=5,\qquad (\mathbb C^4,V(f_4),0)\cong(\mathbb C^4,V(f_5),0) \] by a homeomorphism germ of ambient pairs. The multiplicity $\operatorname{ord}_0f$ is the least total degree appearing among nonvanishing terms of its power series. The equivalence is embedded topology, despite distinct multiplicities.
Mathematical claim 2.
For some finite integer $N>3$ divisible by eight, OpenAI claims reduced real-coefficient polynomials $f_1,f_2\in\mathbb R[z_1,\ldots,z_N]$, each weighted homogeneous for positive rational weights and with an isolated critical point at zero, satisfying \[ \operatorname{ord}_0f_1=2,\qquad\operatorname{ord}_0f_2=3,\qquad (\mathbb C^N,V(f_1),0)\cong(\mathbb C^N,V(f_2),0) \] by an ambient-pair homeomorphism germ. Weighted homogeneous means $f(t^{a_1}z_1,\ldots,t^{a_N}z_N)=t^df(z)$ for positive integers $a_i,d$. The existence assertion does not specify a fixed small ambient dimension.

Outline of the argument

Manuscript 1 (family 059, 2026-09-27): Convergent equations in analytic charts give a finite-map lattice calculation that matches the integral Seifert forms. The nonspherical open-book classification on the seven-sphere identifies the embedded links, and the ambient cone construction transfers that identification to the zero-set germs.

Manuscript 2 (family 059, 2026-09-24): A finite integer relation builds chain-polynomial seeds with matching spectra but different lowest degrees. Tensor operations and local--global isometries align their integral Seifert forms, with determinant cancellation enabling strong approximation; polynomial sums realize the operations, and knot classification extends the resulting link equivalence to the germs.

Formal verification

Family 059 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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