# Cubic-homogeneous counterexamples

- kind: counterexample
- year: 2026
- authors: W. Thompson (24 variables); A. MacFarlane (20); T. Prellberg (19, from Van Rijn's 12-variable map)
- in one line: Explicit non-invertible Keller maps x + H, H cubic, in 24, 20 and 19 variables

**Statement.** There are explicit maps $X+H$ with $H$ homogeneous cubic, $JH$ nilpotent (so $\det\equiv1$), and two distinct rational points with the same image, in $\mathbb C^{24}$, $\mathbb C^{20}$ and $\mathbb C^{19}$.

These maps [refute the cubic-homogeneous Jacobian conjecture explicitly, not only by the Bass–Connell–Wright equivalence](/wiki/Statement:cubic-homogeneous-jc.md).

- On 21 July 2026 Thompson ran [the Bass–Connell–Wright reduction](/wiki/Statement:bcw-cubic-homogeneous.md) on Alpöge's map by hand. The result has 24 variables, 54 cubic monomials, identity linear part, determinant 1 and an explicit rational collision, with two independent exact verifiers. It claims neither priority nor minimality.[1]
- MacFarlane restricted Thompson's map to 20 variables.
- Prellberg works with a 19-variable map, the homogenization of Van Rijn's 12-variable map of degree 3, which keeps a collision.[2]

None of these has been re-run by the atlas, hence V4. Through the symmetric construction, each gives a gradient counterexample in twice the dimension: see [gradient counterexamples](/wiki/Statement:gradient-keller-counterexamples.md). Through Derksen–van den Essen–Zhao, a counterexample in $r$ variables makes the Gaussian moments conjecture fail in $2r$. Long's direct examples in three variables are far smaller.

No explicit counterexample of Drużkowski's cubic-linear form has been reported in the sources the research consulted.

## Relations

- refutes [cubic-homogeneous-jc](/wiki/Statement:cubic-homogeneous-jc.md) — These maps refute the cubic-homogeneous Jacobian conjecture explicitly, not only by the Bass–Connell–Wright equivalence.
- uses [bcw-cubic-homogeneous](/wiki/Statement:bcw-cubic-homogeneous.md) — On 21 July 2026 Thompson ran the Bass–Connell–Wright reduction on Alpöge's map by hand.

## References

1. W. Thompson, [An Explicit 24-Variable Cubic-Homogeneous Reduction of the Alpöge–Fable Jacobian Counterexample](https://zenodo.org/doi/10.5281/zenodo.21466221), Zenodo software v0.1.0, 21 July 2026.
2. T. Prellberg, [Collision-Generated Compression for Homogeneous Keller Maps](https://arxiv.org/abs/2608.12543), arXiv:2608.12543 (12 August 2026; revised 15 September 2026).

## Relations to this page

- related to [gradient-keller-counterexamples](/wiki/Statement:gradient-keller-counterexamples.md) — They come from the cubic-homogeneous counterexamples by doubling.

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