Definition
The Jacobian conjecture (Keller, 1939) asked whether a polynomial map ℝⁿ/ℂⁿ → ℝⁿ/ℂⁿ with everywhere-nonzero constant Jacobian determinant must be globally invertible via a polynomial inverse. It appeared on Smale’s list of 21st-century problems.
Key Points
- 2026-07-19/20: levent-alpoge announced a short C³→C³ counterexample, crediting claude-fable-5; community verification reported as straightforward algebra
- Counterexample: constant Jacobian determinant −2; three distinct inputs map to one output — not invertible
- Disproves the conjecture in three variables; the two-variable case may still be open
- Historically notorious for false published “proofs”
Warning
Formal write-up / arXiv peer review may still be pending. Frame as announced, community-checkable counterexample — not settled journal literature.
Related
- ai-assisted-research
- ai-for-science
- automated-theorem-proving
- ai-mathematical-discovery
- claude-fable-5
- levent-alpoge
- formal-verification