A constant determinant with three inputs sharing an output

Statement

Set $s=xy$ and $u=1+s$ . The map $F(x,y,z)=\begin{pmatrix}u^3z+y^2u(4+3s)\\ y+3xu^2z+3xy^2(4+3s)\\2x-3x^2y-x^3z\end{pmatrix}$ has $\det JF=-2$ . The distinct inputs (0,0,−1/4), (1,−3/2,13/2) and (−1,3/2,13/2) all map to (−1/4,0,0).

Assumptions

Use the displayed map over a characteristic-zero field.

Scope

A concrete failure of injectivity and hence of a polynomial inverse. The coordinate degrees are 7, 6 and 4; a cubic fiber relation does not make this a cubic coordinate map.

Explanation

Constructions · 1

Exact polynomial map and repeated fiber

Three variables over the rationals, also viewed over the complexes. This witness does not settle the two-variable question.

A constant determinant with three inputs sharing an output →

Evidence

Source reports

Proved

Formal polynomial counterexample reported complete ↗

The pinned algebraic statements, not an analytic derivative identification.

Alpöge counterexample in Lean · README trust boundary; Alpoge/Basic.lean and Alpoge/FormalConjectures.lean →

Observed Oct 13, 2026

Proved

Algebraic witness analyzed ↗

The explicit non-injective map.

Counterexamples to the Jacobian conjecture in dimensions greater than two · Introduction and analysis of the counterexample →

Observed Oct 13, 2026

Replaces an earlier source assessment. Version history ↓

Formal proofs

Lean: the three-point fiber ↗

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

Declaration & revision

Alpoge.three_point_fiber

Commit b8eec94711a9b73151929b78cbd7cb999a008590

Connections

In this problem · publication v8 · Selected context

Jacobian conjecture →

Counterexample · Proved

Related statements

This statement: A constant determinant with three inputs sharing an output

The three-variable witness settles the universal formulation, while the two-variable restriction remains a different question.

A collision in ℂ³ supplies no counterexample in ℂ².

Details & sources

A constant determinant with three inputs sharing an output →

Proved · version 3

Set $s=xy$ and $u=1+s$ . The map $F(x,y,z)=\begin{pmatrix}u^3z+y^2u(4+3s)\\ y+3xu^2z+3xy^2(4+3s)\\2x-3x^2y-x^3z\end{pmatrix}$ has $\det JF=-2$ . The distinct inputs (0,0,−1/4), (1,−3/2,13/2) and (−1,3/2,13/2) all map to (−1/4,0,0).

Use the displayed map over a characteristic-zero field.

Scope: A concrete failure of injectivity and hence of a polynomial inverse. The coordinate degrees are 7, 6 and 4; a cubic fiber relation does not make this a cubic coordinate map.

The two-variable Jacobian question →

Open · version 2

Every polynomial map F: $\mathbb C^2$ → $\mathbb C^2$ with constant nonzero Jacobian determinant has a polynomial inverse.

Exactly two complex variables.

Scope: The source leaves this dimension open; the three-variable witness cannot be specialized to refute it.

Counterexamples to the Jacobian conjecture in dimensions greater than two →

Counterexamples to the Jacobian conjecture in dimensions greater than two →

Relationship not independently reviewed.

Connection in Jacobian conjecture →

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.

Alpöge counterexample in Lean →

README trust boundary; Alpoge/Basic.lean and Alpoge/FormalConjectures.lean → · Original passage ↗
Recorded wording
The source proves constant formal polynomial Jacobian determinant, a three-point fiber, and failure of a polynomial inverse for the recorded Jacobian conjecture statement.
paraphrase · OpenMath supplies original paraphrases and links; no proof text is republished.

alerad/alpoge-lean project · Project documentation · Publication date Jul 23, 2026

Public formal source; build and independent mathematical review are separate checks.

Alpöge counterexample in Lean — pinned source ↗No OpenMath execution recorded

Public source inspected for statement and assumptions; this proof project has not been executed by OpenMath.

Revision & attribution

Source revision: b8eec94711a9b73151929b78cbd7cb999a008590. Observed Oct 13, 2026.

The project reports an AI-assisted formalization and computer-algebra workflow; this is distinct from the original discovery.

Source updated Jul 23, 2026.

Alpöge counterexample in Lean — pinned source: b8eec94711a9b73151929b78cbd7cb999a008590.

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 · 2

Assessments retained in these publication records. Earlier statement versions retain their own source histories.

Proved

Algebraic witness analyzed ↗

The explicit non-injective map.

Observed Oct 13, 2026 · Source passage →

Replaces an earlier source assessment; see the original source for its history.

Proved

Formal polynomial counterexample reported complete ↗

The pinned algebraic statements, not an analytic derivative identification.

Observed Oct 13, 2026 · Source passage →

Record details

Claim ID: claim-jacobian-three-point-fiber
Statement version: 3
SHA-256: 60118fc16754df0820d54497ed5053c9f2d67482117106a84f9fc88f42c91137

Publication records

Jacobian conjecture · publication v8 · JSON →