The Jacobian conjecture, which states that a polynomial map in complex variables with a non-zero constant Jacobian is invertible, has been proven false in three dimensions. This means there exists a polynomial map that is locally invertible but not globally invertible.

Recently, a counterexample was found using Fable AI. The polynomial provided is explicit, with a degree of seven. While the verification is brief, the construction appears miraculous due to massive cancellations. The example has since been explained geometrically, focusing on local injectivity and affine varieties.

The reformulated counterexample involves an affine variety isomorphic to and a polynomial map that is locally but not globally injective. This is achieved by constructing a multiplication map from to . The map's inherent symmetries, particularly scaling, mean it cannot be injective. By imposing a normalization condition using the resultant, the map from a four-dimensional variety to a four-dimensional space is still not globally injective, satisfying the local injectivity requirement.