The non-properness set

Jelonek's theorem locates where a counterexample must fail.^[1] A Keller map is étale, so two preimages can never merge. It can fail to be injective only by preimages escaping to infinity, and that happens exactly over $S_F$. If $S_F$ is empty, the map is proper, and a proper Keller map is an automorphism. So a Keller map is invertible exactly when its non-properness set is empty. That criterion is a classical consequence of the covering argument, not part of Jelonek's theorem.

For every counterexample since July 2026, $S_F$ is a nonempty uniruled hypersurface. For Alpöge's map it is a surface in ℂ³, and over it one of the three preimages goes to infinity. Gao describes his tangent-sweep counterexamples as étale of degree $d>1$ onto an open dense image, with the fiber count dropping only across this hypersurface.^[2] The Fong–Fable theorems register calls the counterexamples open-system phenomena: each one's fold is exiled through the non-properness set.^[3]

1. ↑ Z. Jelonek, The set of points at which a polynomial map is not proper, Ann. Polon. Math. 58 (1993) 259–266.
2. ↑ S. Gao, Counterexamples to the Jacobian conjecture in dimensions greater than two, arXiv:2608.00222.
3. ↑ D. Fong and Claude Fable 5, theorems register (theorem 3), jacobianconjectures.com, 21 July 2026.