Difference between revisions
stabilization-question
Status and infobox
- status:
—→ open - kind:
—→ question - year:
—→ 2026 - statement:
—→ Is there a non-automorphic Keller map of $\mathbb C^{n+1}$ that is not equivalent, under polynomial automorphisms of source and target, to $\Phi\times\mathrm{id}$ for a Keller map $\Phi$ of $\mathbb C^n$? - short:
—→ Is every Keller counterexample in ℂⁿ⁺¹ a padded one from ℂⁿ, up to coordinates?
Relations
- + related to tangent-sweep: This question is left open by Gao's tangent-sweep paper for its own four-dimensional maps.
Source +18 −0
+---
+title: Are there native counterexamples in higher dimensions?
+short: Is every Keller counterexample in ℂⁿ⁺¹ a padded one from ℂⁿ, up to coordinates?
+kind: question
+status: open
+year: 2026
+statement: Is there a non-automorphic Keller map of $\mathbb C^{n+1}$ that is not equivalent, under polynomial automorphisms of source and target, to $\Phi\times\mathrm{id}$ for a Keller map $\Phi$ of $\mathbb C^n$?
+---
+Padding makes every three-dimensional counterexample a four-dimensional one. The question is whether dimension four and up has anything new.
+Two candidates are open.
+
+- [[Statement:harish-k4|Harish's map]] of ℂ⁴ has fiber degree 8, which rules out a padded Alpöge map, of degree 3. It does not rule out a padded three-dimensional map of fiber degree 8, and Gallagher's $F_8$ is one.
+- Gao's $F_4$ has degree 5. It is proved inequivalent to $\Phi\times\mathrm{id}$ only for $\Phi$ of a different degree, and he leaves the degree-5 case open.<ref>S. Gao, [Counterexamples to the Jacobian conjecture in dimensions greater than two](https://arxiv.org/abs/2608.00222), arXiv:2608.00222, Theorem 4.3 and the remark after it.</ref>
+
+Geometric degree is invariant under the equivalence, so it cannot settle the question alone. A proof would need a finer invariant, such as the non-properness set or the grading, compared across source and target. This question is [[related::Statement:tangent-sweep|left open by Gao's tangent-sweep paper for its own four-dimensional maps]].
+
+<references />
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Revision 279 as rendered
Padding makes every three-dimensional counterexample a four-dimensional one. The question is whether dimension four and up has anything new.
Two candidates are open.
- Harish's map of ℂ⁴ has fiber degree 8, which rules out a padded Alpöge map, of degree 3. It does not rule out a padded three-dimensional map of fiber degree 8, and Gallagher's is one.
- Gao's has degree 5. It is proved inequivalent to only for of a different degree, and he leaves the degree-5 case open.[1]
Geometric degree is invariant under the equivalence, so it cannot settle the question alone. A proof would need a finer invariant, such as the non-properness set or the grading, compared across source and target. This question is left open by Gao's tangent-sweep paper for its own four-dimensional maps.
- ↑ S. Gao, Counterexamples to the Jacobian conjecture in dimensions greater than two, arXiv:2608.00222, Theorem 4.3 and the remark after it.