Difference between revisions
cubic-homogeneous-counterexamples
Status and infobox
- tier:
—→ V4 - kind:
—→ counterexample - year:
—→ 2026 - 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 $\mathbb C^{19}$. - short:
—→ Explicit non-invertible Keller maps x + H, H cubic, in 24, 20 and 19 variables
Relations
- + refutes cubic-homogeneous-jc: These maps refute the cubic-homogeneous Jacobian conjecture explicitly, not only by the Bass–Connell–Wright equivalence.
- + uses bcw-cubic-homogeneous: On 21 July 2026 Thompson ran the Bass–Connell–Wright reduction on Alpöge's map by hand.
Source +20 −0
+---
+title: Cubic-homogeneous counterexamples
+short: Explicit non-invertible Keller maps x + H, H cubic, in 24, 20 and 19 variables
+kind: counterexample
+tier: V4
+year: 2026
+authors: W. Thompson (24 variables); A. MacFarlane (20); T. Prellberg (19, from Van Rijn's 12-variable map)
+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 [[refutes::Statement:cubic-homogeneous-jc|refute the cubic-homogeneous Jacobian conjecture explicitly, not only by the Bass–Connell–Wright equivalence]].
+- On 21 July 2026 Thompson ran [[uses::Statement:bcw-cubic-homogeneous|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.<ref>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.</ref>
+- 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.<ref name="pr">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).</ref>
+
+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 [[Statement:gradient-keller-counterexamples|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.
+
+<references />
+
Revision 242 as rendered
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 variables makes the Gaussian moments conjecture fail in . 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.
- ↑ W. Thompson, An Explicit 24-Variable Cubic-Homogeneous Reduction of the Alpöge–Fable Jacobian Counterexample, Zenodo software v0.1.0, 21 July 2026.
- ↑ T. Prellberg, Collision-Generated Compression for Homogeneous Keller Maps, arXiv:2608.12543 (12 August 2026; revised 15 September 2026).