# The degree spectrum of ℂ³

- kind: theorem
- status: proved (evidence tier V4)
- year: 2026
- authors: A. Gallagher (every degree from 3); the exclusion of degree 2 by Campbell's theorem, recorded in the Fong–Fable ledger (20 July 2026)
- in one line: Étale self-maps of ℂ³ have geometric degree exactly 1, 3, 4, 5, …

**Statement.** The generic fiber cardinalities of étale polynomial self-maps of $\mathbb C^3$ are exactly $\{1\}\cup\{3,4,5,\dots\}$.

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](/wiki/Statement:keller-birational.md). For degree 2, a quadratic extension in characteristic zero is Galois, and [Campbell's theorem makes every Galois Keller map an automorphism](/wiki/Statement:campbell-galois.md). 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](/wiki/Statement:gallagher-families.md). 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](/wiki/Statement:jc-plane.md).

## Relations

- uses [keller-birational](/wiki/Statement:keller-birational.md) — Degrees 1 and 2 force an automorphism. Degree 1 is Keller's birational theorem.
- uses [campbell-galois](/wiki/Statement:campbell-galois.md) — For degree 2, a quadratic extension in characteristic zero is Galois, and Campbell's theorem makes every Galois Keller map an automorphism.
- uses [gallagher-families](/wiki/Statement:gallagher-families.md) — Every degree from 3 occurs. Gallagher's family has a counterexample of every geometric degree n ≥ 3.

## References

1. S. Gao, [Counterexamples to the Jacobian conjecture in dimensions greater than two](https://arxiv.org/abs/2608.00222), arXiv:2608.00222.
2. D. Fong and Claude Fable 5, [claims ledger](https://jacobianconjectures.com/CLAIMS.md), corrections of 20 July 2026.

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