# Xavier's dichotomy for the Keller locus

- kind: theorem
- status: proved (evidence tier V4)
- year: 2026
- authors: Frederico Xavier (arXiv:2606.22041, 20 June 2026)
- in one line: A component of {deg ≤ k, det = 1}: all automorphisms, or generically none

**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.[1]

- 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 [the Jacobian conjecture as the claim that every component of every bounded-degree Keller locus lies in the automorphism group](/wiki/Statement:jc-n.md). After July 2026 the second alternative occurs. Alpöge's map, rescaled to determinant 1, lies in $V_3^7$. So [the component through Alpöge's map is one whose general member is a counterexample](/wiki/Statement:alpoge-counterexample.md). The Fong–Fable ledger cited the paper on 20 July as prior art for its family and moduli framing.

## Relations

- related to [jc-n](/wiki/Statement:jc-n.md) — 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](/wiki/Statement:alpoge-counterexample.md) — So the component through Alpöge's map is one whose general member is a counterexample.

## References

1. 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.

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