Theorem · version 1
For every integer n ≥ 3 there is a polynomial map F: ℂⁿ → ℂⁿ whose Jacobian determinant is a nonzero constant and which has no polynomial inverse.
Extends the three-variable failure to every higher dimension. Shown as proposed: the preprint has no recorded formal or independent check beyond its own exact computations.
Disproved
Only the exact statement and passage cited here; not an independent OpenMath assessment.
Counterexamples to the Jacobian conjecture in dimensions greater than two · Abstract →
Source dated Jul 31, 2026 · Observed Oct 21, 2026
In this problem · publication v8
Counterexample · Proposed
A family of counterexamples in every dimension above two; shown as proposed until checked beyond the preprint.
Relationship not independently reviewed. This concerns its use in this problem.
Gao generalizes the tangent-sweep mechanism behind the three-variable counterexample and states that it produces counterexamples in every dimension greater than two, with explicit maps in dimensions three, four and five.
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.
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.
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Disproved
Only the exact statement and passage cited here; not an independent OpenMath assessment.
Source dated Jul 31, 2026 · Observed Oct 21, 2026 · Source passage →