Difference between revisions
degree-spectrum-c3
Status and infobox
- status:
—→ proved - tier:
—→ V4 - kind:
—→ theorem - year:
—→ 2026 - statement:
—→ The generic fiber cardinalities of étale polynomial self-maps of $\mathbb C^3$ are exactly $\{1\}\cup\{3,4,5,\dots\}$. - short:
—→ Étale self-maps of ℂ³ have geometric degree exactly 1, 3, 4, 5, …
Relations
- + uses keller-birational: Degrees 1 and 2 force an automorphism. Degree 1 is Keller's birational theorem.
- + uses campbell-galois: 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: Every degree from 3 occurs. Gallagher's family has a counterexample of every geometric degree n ≥ 3.
Source +22 −0
+---
+title: The degree spectrum of ℂ³
+short: Étale self-maps of ℂ³ have geometric degree exactly 1, 3, 4, 5, …
+kind: theorem
+status: proved
+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)
+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 [[uses::Statement:keller-birational|Keller's birational theorem]]. For degree 2, a quadratic extension in characteristic zero is Galois, and [[uses::Statement:campbell-galois|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.** [[uses::Statement:gallagher-families|Gallagher's family has a counterexample of every geometric degree n ≥ 3]]. Gao's tangent sweeps give the same range in every dimension $n\ge3$.<ref>S. Gao, [Counterexamples to the Jacobian conjecture in dimensions greater than two](https://arxiv.org/abs/2608.00222), arXiv:2608.00222.</ref>
+
+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.<ref>D. Fong and Claude Fable 5, [claims ledger](https://jacobianconjectures.com/CLAIMS.md), corrections of 20 July 2026.</ref> 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 [[Statement:jc-plane|JC₂]].
+
+<references />
+
Revision 245 as rendered
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 .
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 .[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 . What is excluded there is drawn on JC₂.
- ↑ S. Gao, Counterexamples to the Jacobian conjecture in dimensions greater than two, arXiv:2608.00222.
- ↑ D. Fong and Claude Fable 5, claims ledger, corrections of 20 July 2026.