[#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.
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
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]
1Records
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
- Set-theoretic complete intersections for complex space curvesalgebraic geometry
- A polynomial bijection from the rational plane to the rational linealgebraic geometry
- Hodge conjecturealgebraic geometry
How to cite
TheoremDB contributors, “Plane Jacobian conjecture,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/plane-jacobian-conjectureThis page as plain text: plane-jacobian-conjecture.md
This problem includes 2 records joined by 2 typed links, sourced from mathworld.wolfram.com[1], current as of July 31, 2026.
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.