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Jacobian Conjecture

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jacobian-conjectureAlgebraposed by Ott-Heinrich Keller, 1939recorded: disproved

1 attempt · 1 machine check · no person has looked

Statement

Every polynomial map Cn→Cn\mathbb{C}^n \to \mathbb{C}^n with constant nonzero Jacobian determinant is invertible, with a polynomial inverse.

Context

n ≥ 3; plane case open

On Smale's problem list and notorious across algebraic geometry since 1939.

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no project yet · nobody looking

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Interest

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Attempts

1 attempt

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  • #1

    Attempt 1

    constructionClaude Fable 5 with Levent Alpöge ·
    AI involvement
    ai discovered
    — the result was found by a model.
    models
    Claude Fable 5
    people
    Levent Alpöge

    Alpöge used Fable 5 as a research collaborator to hunt down an explicit counterexample map in three variables, rather than running a generic formal-proof search. The two-variable (plane) case of the conjecture remains open.

    n ≥ 3; plane case open

    Reviews

    0 human reviews · 1 machine check

    No person has reviewed this attempt. 1 machine check below — a machine check is not human verification.

    • Machine check · not human verification

      machine: correct

      Recorded from VibeMathed expert verification (imported) ·

      scope Expert review reported upstream; the reviewer has no probXiv account and is not credited here

      The counterexample is hand-checkable by direct substitution and was independently confirmed by outside mathematicians within hours of posting. Briefly caught in a Wikipedia edit war over whether to record it. No formal peer-reviewed publication yet.

      No ProbXiv account is credited for this check. Nobody has put their name to it, so it carries no personal accountability and does not count as verification by a person.

    Endorsements

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    Discussion of this attempt

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Discussion

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