Counterexamples to the Jacobian conjecture in dimensions greater than two

Abstract

The Jacobian conjecture, open since 1939, asks whether every polynomial map of $\C^n$ whose Jacobian determinant is a nonzero constant must have a polynomial inverse. It was refuted in dimension three by Alp\"oge on July 19, 2026, with an infinite family by Gallagher (July 20) and a geometric explanation by Speyer (July 23): the counterexample sweeps the tangent lines of a plane curve --- a map that classical duality forces to hit most points several times. We give a self-contained account of this tangent-sweep mechanism and generalize it from plane curves to direction fields on hypersurfaces. The resulting construction produces counterexamples in every dimension greater than two and, in each dimension, of arbitrarily large geometric degree (the number of preimages of a typical point). We work it out in five new explicit maps: one three-dimensional of degree four, two four-dimensional of degrees five and ten, and two five-dimensional of degrees six and twelve. The counterexamples provide explicit examples of \'etale coverings $\C^n\to\C^n$ that are not proper: they are everywhere unramified, and fail to be injective only through points escaping to infinity. All identities were verified in exact rational arithmetic; an appendix determines exact fiber structures through Gr\"obner bases.