Difference between revisions
keller-locus-dichotomy
Status and infobox
- status:
—→ proved - tier:
—→ V4 - kind:
—→ theorem - year:
—→ 2026 - statement:
—→ Let $V_n^k$ be the variety of polynomial maps $F:\mathbb C^n\to\mathbb C^n$ with $\deg F\le k$ and $\det DF\equiv1$, and $\Omega$ an irreducible component. Either $\Omega\subset\mathrm{Aut}(\mathbb C^n)$, or a dense Zariski-open subset of $\Omega$ consists of maps that are not automorphisms. - short:
—→ A component of {deg ≤ k, det = 1}: all automorphisms, or generically none
Relations
- + related to jc-n: The theorem recasts the Jacobian conjecture as the claim that every component of every bounded-degree Keller locus lies in the automorphism group.
- + related to alpoge-counterexample: So the component through Alpöge's map is one whose general member is a counterexample.
Source +20 −0
+---
+title: Xavier's dichotomy for the Keller locus
+short: A component of {deg ≤ k, det = 1}: all automorphisms, or generically none
+kind: theorem
+status: proved
+tier: V4
+year: 2026
+authors: Frederico Xavier (arXiv:2606.22041, 20 June 2026)
+statement: Let $V_n^k$ be the variety of polynomial maps $F:\mathbb C^n\to\mathbb C^n$ with $\deg F\le k$ and $\det DF\equiv1$, and $\Omega$ an irreducible component. Either $\Omega\subset\mathrm{Aut}(\mathbb C^n)$, or a dense Zariski-open subset of $\Omega$ consists of maps that are not automorphisms.
+---
+Xavier's paper appeared a month before the counterexample, and it says nothing about which alternative occurs. The proof uses four facts.<ref>F. Xavier, [An organizing principle in the study of the Jacobian Conjecture](https://arxiv.org/abs/2606.22041), arXiv:2606.22041 (20 June 2026). Unrefereed.</ref>
+- The non-injective maps in the parameter space form a constructible set (Chevalley).
+- The injective members of $\Omega$ are closed in the Euclidean topology, because a limit of injective local homeomorphisms is injective.
+- A constructible set that is Euclidean-closed is Zariski-closed.
+- Injective polynomial self-maps of ℂⁿ are automorphisms.
+
+The theorem recasts [[related::Statement:jc-n|the Jacobian conjecture as the claim that every component of every bounded-degree Keller locus lies in the automorphism group]]. After July 2026 the second alternative occurs. Alpöge's map, rescaled to determinant 1, lies in $V_3^7$. So [[related::Statement:alpoge-counterexample|the component through Alpöge's map is one whose general member is a counterexample]]. The Fong–Fable ledger cited the paper on 20 July as prior art for its family and moduli framing.
+
+<references />
+
Revision 268 as rendered
Xavier's paper appeared a month before the counterexample, and it says nothing about which alternative occurs. The proof uses four facts.[1]
- The non-injective maps in the parameter space form a constructible set (Chevalley).
- The injective members of are closed in the Euclidean topology, because a limit of injective local homeomorphisms is injective.
- A constructible set that is Euclidean-closed is Zariski-closed.
- Injective polynomial self-maps of ℂⁿ are automorphisms.
The theorem recasts the Jacobian conjecture as the claim that every component of every bounded-degree Keller locus lies in the automorphism group. After July 2026 the second alternative occurs. Alpöge's map, rescaled to determinant 1, lies in . So the component through Alpöge's map is one whose general member is a counterexample. The Fong–Fable ledger cited the paper on 20 July as prior art for its family and moduli framing.
- ↑ F. Xavier, An organizing principle in the study of the Jacobian Conjecture, arXiv:2606.22041 (20 June 2026). Unrefereed.