Jacobian conjecture disproved: Fable AI discovers 3D polynomial counterexample

Summary

Fable AI discovered a three-dimensional multivariate map; despite the Jacobian determinant being constantly 1, the map is not invertible, thereby disproving the Jacobian conjecture in three and higher dimensions.

Why it matters

The Jacobian Conjecture is one of the oldest unsolved problems of the twentieth century; this refutation expands the boundaries of mathematical thought while demonstrating AI's potential in deep theoretical discovery. It also opens new research avenues into the behavior of high-dimensional polynomials.

Highlights

• An explicit three-dimensional polynomial example, F(x,y,z) = (x, y + x^2, z + xy + x^3), was presented.
• The Jacobian determinant of this map is constantly 1 at every point, yet the map is globally non-invertible.
• Fable AI's striking discovery proves that artificial intelligence can be an effective tool for uncovering complex mathematical structures.

Critical take

The proof relies on advanced concepts from algebraic geometry and requires independent verification; moreover, the conjecture remains open in two dimensions, raising questions about the general applicability of this method.