John D. Cook • 7/21/2026

Locally everywhere does not imply everywhere

This article discusses how mathematician Levent Alpöge discovered a counterexample to the Jacobian conjecture using Claude Fable 5. The Jacobian conjecture states that a polynomial function from ℝⁿ to ℝⁿ with a constant, non-zero Jacobian has a polynomial inverse. Alpöge found a specific polynomial function from ℝ³ to ℝ³ with constant Jacobian determinant -2 that is locally invertible everywhere but not globally invertible, disproving the conjecture for n=3. The article also notes that the conjecture remains open for n=2 and can be extended to higher dimensions.

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#mathematics #Counterexample #Algebraic Geometry