# Are there native counterexamples in higher dimensions?

- kind: question
- status: open
- year: 2026
- in one line: Is every Keller counterexample in ℂⁿ⁺¹ a padded one from ℂⁿ, up to coordinates?

**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.

- [Harish's map](/wiki/Statement:harish-k4.md) 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.[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](/wiki/Statement:tangent-sweep.md).

## Relations

- related to [tangent-sweep](/wiki/Statement:tangent-sweep.md) — This question is left open by Gao's tangent-sweep paper for its own four-dimensional maps.

## References

1. 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.

## Relations to this page

- related to [harish-k4](/wiki/Statement:harish-k4.md) — So whether this map is genuinely four-dimensional is an instance of the open stabilization question.

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