Cubic-homogeneous counterexamples

These maps refute the cubic-homogeneous Jacobian conjecture explicitly, not only by the Bass–Connell–Wright equivalence.

• On 21 July 2026 Thompson ran the Bass–Connell–Wright reduction 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. 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.

1. ↑ W. Thompson, An Explicit 24-Variable Cubic-Homogeneous Reduction of the Alpöge–Fable Jacobian Counterexample, Zenodo software v0.1.0, 21 July 2026.
2. ↑ T. Prellberg, Collision-Generated Compression for Homogeneous Keller Maps, arXiv:2608.12543 (12 August 2026; revised 15 September 2026).