Arnold's corank problem: is the Hessian corank of a hypersurface singularity an ambient topological invariant?

The weighted homogeneous germs $f_1,f_2$ on $\mathbb{C}^N$ ( $N>3$ , divisible by eight) of the Zariski counterexample are ambiently homeomorphic, have isolated critical points and real coefficients, and have Hessian ranks $16E>0$ and $0$ , so their coranks differ. Corank parity is preserved, as Sampaio's theorem requires. Nothing is said about corank in three variables (where it is invariant) or about a minimal dimension.

Exact FrontierDelta

Prior state unknown→disproved

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Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Arnold's corank problem: is the Hessian corank of a hypersurface singularity an ambient topological invariant? · Arnold's corank problem

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Registry verification: unreviewed · announcement · candidate

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VibeMathed record: Arnold's corank problem: is the Hessian corank of a hypersurface singularity an ambient topological invariant?

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No independent mathematician has checked this yet. Checked here: the corollary 'Negative answer to Arnold's corank problem' in the September 24 manuscript was read against the question as the manuscript states it from Sampaio 2026. It is a corollary of the same pair of germs as the Zariski counterexample, whose Hessian ranks are 16E and 0 for an integer E fixed by the construction. Not refereed; no Lean formalization. The paper notes the result is consistent with Sampaio's corank-parity theorem, since both ranks are even, and with the three-variable invariance theorem, since N > 3.

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The Budur-Fernandez de Bobadilla-Le-Nguyen conjecture on initial-form Milnor fibres of topologically equivalent germs

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Sampaio, On the Arnold corank, the Zariski multiplicity, and the tangent Milnor fiber problems (2026)

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problem-record link supplied by the VibeMathed registry.

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VibeMathed record: Arnold's corank problem: is the Hessian corank of a hypersurface singularity an ambient topological invariant? evidence for this event

The Budur-Fernandez de Bobadilla-Le-Nguyen conjecture on initial-form Milnor fibres of topologically equivalent germs evidence for this event

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Sampaio, On the Arnold corank, the Zariski multiplicity, and the tangent Milnor fiber problems (2026) evidence for this event

Arnold's corank problem: is the Hessian corank of a hypersurface singularity an ambient topological invariant? parent of this event

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