Preliminary mathematical note - version 0.6 - 21 July 2026
An explicit cubic homogeneous counterexample in dimension 19
Starting from the factor-aware 11-variable degree-three map posted in Joe Atkins-Turkish's Gist, a sparse Segre-Yagzhev lift uses only the seven nonzero normalized cubic components to produce a 19-variable Yagzhev-form map.
Exact claim. There is an explicit polynomial map
such that every nonzero component of H is cubic homogeneous,
and three distinct rational points have the same image. Consequently det JG = 1 identically, while G is not injective.
Compression result. Among the timestamped public constructions located as of 21 July 2026, this reduces the smallest dimension of a fully explicit counterexample in cubic homogeneous Yagzhev form from 79 to 19. The 11- and 17-variable public maps are degree-three Keller maps, not yet Yagzhev-form maps.
Status. This is a rapidly prepared preliminary draft about a newly announced counterexample. The displayed 19-dimensional object is independently machine-checkable by finite exact-arithmetic certificates, but the note has not undergone ordinary peer review.
Paper and verification artifacts
The final object is stored in a canonical JSON representation. The SymPy script derives the map from the 11-variable input and checks the global identities; the second script independently checks the same JSON using only Python's standard library.
- Paper, version 0.6 Compiled preliminary draft (PDF)
- Versioned LaTeX source Complete source for the note
- Current-paper alias Stable URL for the current PDF
- Canonical map Machine-readable H, collision points, and image
- SymPy verifier Derivation and exact symbolic certificate
- Standard-library verifier Independent dictionary-polynomial implementation
- Generated appendices Typeset formulas generated and byte-for-byte checked by the SymPy verifier
- README Commands, versions, hashes, and attribution notes
- Versioned release archive Exact version 0.6 bundle
- SHA-256 manifest Hashes for the distributed release files
- Licenses MIT for code; CC BY 4.0 for manuscript and map
Run locally
python3 -m pip install 'sympy==1.14.0'
python3 verify_yagzhev19_sympy.py
python3 verify_yagzhev19_stdlib.py yagzhev19_map.jsonTested with Python 3.13.5 and SymPy 1.14.0.
Canonical JSON SHA-256: f88a0883ce9509339c53a6a721390053b424039a8dd6c41ebc7770aeb955cced
Public constructions and provenance
The note does not claim authorship of the 11-variable input map. It records the public chain of constructions and isolates the sparse 19-dimensional endpoint and its global exact certificate.
| Contributor | Public object | Dimension |
|---|---|---|
| Alpöge / Fable 5 | Original Keller counterexample | 3 |
| Ali Demirci | Generic explicit Yagzhev realization | 79 |
| u/Distinct-Soft-3991 | Degree-three Keller map | 17 |
| Joe Atkins-Turkish | Posted factor-aware degree-three Keller map; Gist credits ChatGPT | 11 |
| Present note | Sparse Yagzhev lift with global symbolic certificate | 19 |
Contribution and AI disclosure
Harris Chan formulated the sparse homogenization, directed the symbolic construction and independent verification, audited the generated artifacts, and prepared the exposition, with assistance from Claude Fable 5, GPT-5.6 Sol, and Gemini 3.1 Pro. These systems are acknowledged as computational tools rather than authors; Harris Chan takes responsibility for the mathematical claims, citations, attribution decisions, and conclusions.
Citation
Please cite the versioned PDF and include the access date while the note remains preliminary.
BibTeX
@misc{chan2026yagzhev19,
author = {Harris Chan},
title = {An Explicit Cubic Homogeneous Counterexample
to the Jacobian Conjecture in Dimension 19},
year = {2026},
month = jul,
note = {Preliminary draft, version 0.6},
url = {https://harrischan.com/docs/yagzhev-19/}
}