Disproved
Explicit three-variable counterexample ↗
The all-dimensions complex Jacobian conjecture.
Observed Oct 13, 2026
Replaces an earlier source assessment. Version history ↓
Conjecture · version 3
Constant Jacobian implies a polynomial inverse
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 v3 · Selected context
Jacobian conjecture →
Primary formulation · Disproved
Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v8
Jacobian conjecture →
Primary formulation · Disproved
Relationship not independently reviewed. This concerns its use in this problem.
Sources
A Counterexample to the Jacobian Conjecture →
- 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.
ULAM research manuscript (no author named) · Preprint · Publication date Jul 20, 2026
Public manuscript or exposition; publication does not imply independent OpenMath review.
Revision & attribution
Source revision: 2026-07-20. Observed Oct 13, 2026.
The manuscript cites Levent Alpöge’s announcement, credits Akhil Mathew for asking the question and Fable for the work leading to the example. No named author is printed on this verification manuscript; these discovery credits are not its bibliographic authorship.
Source updated Jul 20, 2026.
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 →
Replaces an earlier source assessment; see the original source for its history.
Record details
- Claim ID
- claim-jacobian-polynomial-inverse
- Statement version
- 3
- SHA-256
- fb842549e2d79f65dea8544847804d9bb650320c9d90ec327390515a3b5a989d