Levent Alpoge announced on Twitter that the Jacobian Conjecture, a longstanding problem in mathematics, has been disproven. The counterexample was developed by his colleague Fable during the World Cup final and involves a polynomial map from \(\mathbb{C}^3\) to \(\mathbb{C}^3\) with a Jacobian determinant of -2, contradicting the conjecture's claim that such maps must have determinant 1.
The explicit polynomial map is given by \(((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z)\). It sends points \((0, 0, -1/4), (1, -3/2, 13/2), (-1, 3/2, 13/2)\) to \((-1/4, 0, 0)\), demonstrating the failure of the conjecture. Alpoge credited his friend Akhil for prompting the inquiry and Fable for the construction.
The Jacobian Conjecture, proposed in 1939, has been a central open problem in algebraic geometry and polynomial mappings, asserting that polynomial maps with constant nonzero Jacobian determinant are invertible with polynomial inverses. This counterexample challenges decades of assumptions and could have implications for related fields such as dynamical systems and complex analysis. The determinant value of -2 directly refutes the conjecture's requirement of determinant 1.
Alpoge's post includes a Wolfram Alpha link verifying the determinant calculation. The announcement was made on July 20, 2026, and has since sparked extensive discussion in mathematical circles online, marking a pivotal moment in the study of polynomial automorphisms.