The degree spectrum of ℂ³

On ℂⁿ an étale polynomial map has a Jacobian determinant that never vanishes. That determinant is a polynomial, so it is constant, and étale maps are exactly Keller maps. The geometric degree of such a map is the number of preimages of a general point, which is the degree of the function field extension $\mathbb C(x)/\mathbb C(F)$.

The theorem has two halves.

• Degrees 1 and 2 force an automorphism. Degree 1 is Keller's birational theorem. For degree 2, a quadratic extension in characteristic zero is Galois, and Campbell's theorem makes every Galois Keller map an automorphism. So no counterexample has degree 2, and 3 is the least possible degree.
• Every degree from 3 occurs. Gallagher's family has a counterexample of every geometric degree n ≥ 3. Gao's tangent sweeps give the same range in every dimension $n\ge3$.^[1]

The ledger's open questions O2 (does degree 4 occur?) and O3 (the whole spectrum) were answered by this on the day they were posed.^[2] The tier is V4 because the upper half rests on unrefereed uniform proofs. Its members from 3 to 100 have exact certificates.

In the plane the spectrum is unknown, and it is the question: JC₂ says it is $\{1\}$. What is excluded there is drawn on JC₂.

1. ↑ S. Gao, Counterexamples to the Jacobian conjecture in dimensions greater than two, arXiv:2608.00222.
2. ↑ D. Fong and Claude Fable 5, claims ledger, corrections of 20 July 2026.