Constant Jacobian implies a polynomial inverse

Statement

For every integer n ≥ 1, every polynomial map F: ℂⁿ → ℂⁿ whose Jacobian determinant is a nonzero constant has a polynomial inverse.

Universal conjecture; one non-injective map in dimension three suffices to refute it.

Disproved

The all-dimensions complex Jacobian conjecture.

Observed Oct 13, 2026

In this problem · publication v1

Primary formulation · Disproved

Relationship not independently reviewed. This concerns its use in this problem.

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.

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.

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

The all-dimensions complex Jacobian conjecture.

Observed Oct 13, 2026 · Source passage →

Record details