Disproved
Explicit three-variable counterexample ↗
The all-dimensions complex Jacobian conjecture.
An explicit Jacobian counterexample · Explicit map and verification of determinant and fiber →
Observed Oct 13, 2026
Conjecture · version 2
Constant Jacobian implies a polynomial inverse
Earlier version 2. View current version 3 →
Statement
For every integer n ≥ 1, every polynomial map F: → whose Jacobian determinant is a nonzero constant has a polynomial inverse.
Assumptions
- The ground field is ℂ.
- The determinant is a nonzero constant polynomial.
Scope
Universal conjecture; one non-injective map in dimension three suffices to refute it.
Evidence
Source reports
Connections
In this problem · publication v2
Jacobian conjecture →
Primary formulation · Disproved
Relationship not independently reviewed. This concerns its use in this problem.
Sources
An explicit Jacobian counterexample →
- Explicit map and verification of determinant and fiber → · Original passage ↗
Recorded wording
An explicit three-variable polynomial map has constant determinant −2 and a fiber containing three distinct rational points.
paraphrase · OpenMath supplies original paraphrases and links; no proof text is republished.
Levent Alpöge · Preprint · Publication date Jul 2026
Public manuscript or exposition; publication does not imply independent OpenMath review.
Revision & attribution
Source revision: 2026-07. Observed Oct 13, 2026.
Discovery is attributed to Levent Alpöge working with Claude Fable 5; mathematical authorship and the model contribution are separate roles.
Counterexamples to the Jacobian conjecture in dimensions greater than two →
- Introduction and analysis of the counterexample → · Original passage ↗
Recorded wording
Algebraic analysis of the three-variable counterexample, including its non-injectivity; the two-dimensional complex case remains open.
paraphrase · OpenMath supplies original paraphrases and links; no proof text is republished.
Shuhong Gao · Preprint · Publication date Jul 31, 2026
Public manuscript or exposition; publication does not imply independent OpenMath review.
Revision & attribution
Source revision: 2026-07-31. Observed Oct 13, 2026.
No AI contribution is asserted by this source record.
Source updated Jul 31, 2026.
History
Statement versions
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessment history · 1
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Disproved
Explicit three-variable counterexample ↗
The all-dimensions complex Jacobian conjecture.
Observed Oct 13, 2026 · Source passage →
Record details
- Claim ID
- claim-jacobian-polynomial-inverse
- Statement version
- 2
- SHA-256
- 38d201aee19fc6fefb68d3b838ca6b5d47e645b95031cee645ea39d3d49cdd35