Jacobian Conjecture
Every polynomial map with constant nonzero Jacobian determinant is invertible, with a polynomial inverse.
- Result
- Disproved(see note)
- Status
- Resolved
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Algebraic Geometry
- Posed by
- Ott-Heinrich Keller
- Year posed
- 1939
- Years open
- 87y
- Solved
- 2026-07-20
- Model
- Claude Fable 5
- Vendor
- Anthropic
- Collaborators
- Levent Alpöge
- Verification
- Independently expert-verified
- Publication
- Announced
- Significance
- 65 / 100
- Disclosed cost
- —
- Wikipedia
- 13 languages
What was actually shown
n ≥ 3; plane case open
What the AI did
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.
Verification
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.