[#P24] Plane Jacobian conjecture

Problem. Let \(F:\mathbb{C}^2\to\mathbb{C}^2\) be a polynomial map. If the Jacobian determinant \(\det J_F\) is a nonzero constant, then \(F\) has a polynomial inverse.

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Definitions and notation

1Context

The broad all-dimensional conjecture was disproved in July 2026. Its original two-variable case remains open.

2Problem setup

Definition 1 (The Jacobian determinant). The Jacobian determinant is the determinant of the matrix of first partial derivatives of the two coordinate polynomials of F.

Definition 2 (A polynomial inverse). A polynomial inverse is a polynomial map G with G composed with F and F composed with G both equal to the identity map.

Remark 1. The broad all-dimensional conjecture was disproved in July 2026. Its original two-variable case remains open.

3What counts as a solution

Prove polynomial invertibility for every polynomial map from C^2 to C^2 with nonzero constant Jacobian determinant, or give such a plane map and rigorously verify that it has no polynomial inverse.

1Status

What counts as a solution

Current status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: The complex conjecture in every dimension reduces to cubic homogeneous maps. Dimension one is elementary, and several restricted plane cases are proved. Exact unresolved remainder: For every polynomial F:C^2 to C^2 with constant nonzero Jacobian determinant, prove that F has a polynomial inverse. Higher-dimensional complex cases remain open as well.^[2][1]

1Packet records

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Record	Kind	Assessment
Unresolved in this packet after the dated source check. Strongest checked result: The complex conjecture in every dimension reduces to cubic homogeneous maps. Dimension one is elementary, and several restricted plane cases are proved. Exact unresolved remainder: For every polynomial F:C^2 to C^2 with constant nonzero Jacobian determinant, prove that F has a polynomial inverse. Higher-dimensional complex cases remain open as well.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 Unresolved in this packet after the dated source check. Strongest checked result: The complex conjecture in every dimension reduces to cubic homogeneous maps. Dimension one is elementary, and several restricted plane cases are proved. Exact unresolved remainder: For every polynomial F:C^2 to C^2 with constant nonzero Jacobian determinant, prove that F has a polynomial inverse. Higher-dimensional complex cases remain open as well.^[2][1] Relevance to this problem For Plane Jacobian conjecture, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 2, Reference 1 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. Strongest checked result: The complex conjecture in every dimension reduces to cubic homogeneous maps. Dimension one is elementary, and several restricted plane cases are proved. Exact unresolved remainder: For every polynomial F:C^2 to C^2 with constant nonzero Jacobian determinant, prove that F has a polynomial inverse. Higher-dimensional complex cases remain open as well. Open full record →
Prove polynomial invertibility for every polynomial map from C^2 to C^2 with nonzero constant Jacobian determinant, or give such a plane map and rigorously verify that it has no polynomial inverse.	Trace	Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Prove polynomial invertibility for every polynomial map from C^2 to C^2 with nonzero constant Jacobian determinant, or give such a plane map and rigorously verify that it has no polynomial inverse.^[1] Relevance to this problem For Plane Jacobian conjecture, record plane-jacobian-conjecture-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Prove polynomial invertibility for every polynomial map from C^2 to C^2 with nonzero constant Jacobian determinant, or give such a plane map and rigorously verify that it has no polynomial inverse. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Prove polynomial invertibility for every polynomial map from C^2 to C^2 with nonzero constant Jacobian determinant, or give such a plane map and rigorously verify that it has no polynomial inverse. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
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Notes and companion material

Original intake status. The cited current reference reports that the general conjecture is false in dimension three and higher while the two-dimensional case remains open. The source and public status were checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.

An explicit three-dimensional counterexample was announced on 2026-07-19 and has direct symbolic checks. This record is restricted to complex dimension two.

Recorded example 1. Every invertible affine-linear plane map satisfies the condition and has a polynomial inverse.

Computational notes

Symbolic calculations can verify bounded degrees or structured families without settling all polynomial plane maps.

2See also

A polynomial bijection from the rational plane to the rational linealgebraic geometry
Set-theoretic complete intersections for complex space curvesalgebraic geometry
Hodge conjecturealgebraic geometry

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Plain text

“Plane Jacobian conjecture.” TheoremDB. P24. Problem statement; statement text SHA-256 7a8e074bfdee1b7050419d5ede8ae629347a3cc77da651338e8e97c8532d5e23. https://theoremdb.org/statement/?ref=P24

BibTeX

@misc{theoremdb-problem-7a8e074bfdee1b7050419d5ede8ae629347a3cc77da651338e8e97c8532d5e23,
  title = {{Plane Jacobian conjecture}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 7a8e074bfdee1b7050419d5ede8ae629347a3cc77da651338e8e97c8532d5e23},
  url = {https://theoremdb.org/statement/?ref=P24}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from mathworld.wolfram.com^[1], current as of July 31, 2026.

1Lean verification

1References

Packet source. Eric W. Weisstein, Jacobian Conjecture, MathWorld. mathworld.wolfram.com checked 2026-08-01. Wolfram MathWorld, current status and dimension-three counterexample note, checked 2026-07-22. ↗website · primary source · checked 2026-07-31Source use: original summary.The cited current reference reports that the general conjecture is false in dimension three and higher while the two-dimensional case remains open. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.Also cited at Problem statement and variants.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.Packet-linked source whose variant wording requires correction.Source named by the research packet.
Hyman Bass, Edwin H. Connell, and David Wright, “The Jacobian conjecture: Reduction of degree and formal expansion of the inverse”. Bulletin of the American Mathematical Society 7(2) (1982), 287-330. DOI 10.1090/S0273-0979-1982-15032-7. Main reduction theorem. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Primary source for the cubic-homogeneous reduction.

An original CC0 restatement prepared by TheoremDB maintainers.

[#P24] Plane Jacobian conjecture

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