Two-dimensional Jacobian Conjecture

Determine whether every complex polynomial mapping F: ℂ² → ℂ² whose Jacobian determinant is nowhere zero has a polynomial inverse, thereby resolving the two-dimensional case of the Jacobian Conjecture.

Background

The Jacobian Conjecture asks whether a complex polynomial mapping with nowhere-vanishing Jacobian determinant must be a polynomial automorphism. The paper explains that the one-dimensional case is elementary and that a counterexample is claimed for dimensions n ≥ 3 via Levent Alpöge’s example and its liftings. Consequently, the unresolved case explicitly identified by the paper is n = 2.

The paper’s main theorem establishes the Jacobian-conjecture implication on a non-empty dense Zariski open subset of the space of polynomial mappings with prescribed bounded degrees. It therefore does not settle whether the implication holds for every polynomial mapping in two variables; instead, it concludes that any two-dimensional counterexample, if one exists, must be nongeneric in the Zariski-topological sense.