Reading source record 2026-07-31. View default cited record →
Algebraic analysis of the three-variable counterexample, including its non-injectivity; the two-dimensional complex case remains open.
Public manuscript or exposition; publication does not imply independent OpenMath review.
The exact passages cited in OpenMath. Each passage identifies how its wording was recorded.
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.
Passage ID: jacobian-conjecture:entry:all-dimensions:v1
Source item: abstract
Source record: jacobian-conjecture:source:analysis:v2
Algebraic analysis of the three-variable counterexample, including its non-injectivity; the two-dimensional complex case remains open.
Replaces an earlier passage. View versioned records →
Passage ID: jacobian-conjecture:entry:analysis:v2
Source item: analysis
Source record: jacobian-conjecture:source:analysis:v2
Replaces passage: jacobian-conjecture:entry:analysis:v1
Explicit citations in versioned problem publications. A citation records a use; it does not confer proof or independent verification.
Problem publication v8
Source assessment
Disproved
Counterexamples in every dimension n ≥ 3
Only the exact statement and passage cited here; not an independent OpenMath assessment.
Source date Jul 31, 2026 · Observed Oct 21, 2026
Source assessment
Open
Dimension two remains open
Complex dimension two.
Observed Oct 13, 2026
Source assessment
Proved
Algebraic witness analyzed
The explicit non-injective map.
Observed Oct 13, 2026
Construction
Exact polynomial map and repeated fiber
Formal declaration
Lean: the three-point fiber
Question
Jacobian conjecture
Connection
The three-variable witness settles the universal formulation, while the two-variable restriction remains a different question.
History
Keller’s global invertibility question
History
Separate algebraic analysis
Further reading
Read the setting and notation before comparing the results.
Linked manuscripts, code and data. External proof reports, statement correspondence and OpenMath execution are separate checks.
Complete formal proof reported externally
The polynomial witness is source-reported complete; OpenMath has not built it. The project also proves jacobian_det and not_injective.
Statement match: Linked by OpenMath · independent review separate · checked Oct 13, 2026
Alpoge.three_point_fiberCommit b8eec94711a9b73151929b78cbd7cb999a008590
Recorded authorship
Shuhong Gao
No AI contribution is asserted by this source record.
OpenMath source records preserve exact citations. Observation dates order these records; they do not establish the latest upstream paper or proof.
Counterexamples to the Jacobian conjecture in dimensions greater than two
On a counterexample to the Jacobian conjecture
OpenMath supplies original paraphrases and links; no proof text is republished.
Source terms ↗