A Counterexample to the Weighted Homogeneous Jacobian Jet Closure Conjecture
Manuscript 22 October 2026 · Online 23 October 2026
Abstract
Chen and Zuo conjectured that the rth jet closure of the Jacobian ideal of a weighted homogeneous polynomial equals that ideal plus the (r+1)st power of the maximal ideal. For f=z(x^2+y^3), we exhibit xy^3 in the fourth jet closure but outside J(f)+(x,y,z)^5. Membership follows from an explicit identity in universal jet coefficients, and nonmembership from a homomorphism to k[s]/(s^10). We also give an irreducible four-variable example and a transfer principle for principal-ideal jet defects. The construction uses the published cusp example of de Fernex, Ein and Ishii. Both examples are nonisolated and disprove the conjecture as written, not a version restricted to isolated singularities. The all-orders OWR-4086-008 question is already resolved by Mallory and is not claimed as a new result of this note.
Record
- Affiliation
- Mercury Software GmbH
- Contact
- [email protected] · GitHub
- Result
- Counterexample to an adjacent jet-closure conjecture
- Categories
- math.AC · math.AG
- Manuscript
- 22 October 2026
- Online release
- 23 October 2026
- Version
- 1.0
- License
- Creative Commons Attribution 4.0 International
Files and verification
The PDF is the canonical reading copy. The source archive contains the LaTeX manuscript, bibliography, reproducibility material, and audit documents without build artefacts.
Citation
Alper Ferudun, “A Counterexample to the Weighted Homogeneous Jacobian Jet Closure Conjecture,” EulerSolve Research Papers, OWR-4086-008, 2026. https://doi.org/10.5281/zenodo.23587224.
BibTeX
@misc{ferudun2026owr4086008,
author = {Ferudun, Alper},
title = {A Counterexample to the Weighted Homogeneous Jacobian Jet Closure Conjecture},
year = {2026},
howpublished = {EulerSolve Research Papers},
url = {https://eulersolve.org/papers/owr-4086-008/},
doi = {10.5281/zenodo.23587224},
note = {OWR-4086-008; unrefereed preprint}
}