# P24: Plane Jacobian conjecture

- ID: `P24`
- Reference: `plane-jacobian-conjecture`
- Page: https://theoremdb.org/statements/P24
- Record maturity: Reviewed problem with recorded work

## 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.

### Context

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

### Problem setup

- **Definition (The Jacobian determinant).** The Jacobian determinant is the determinant of the matrix of first partial derivatives of the two coordinate polynomials of F.
- **Definition (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.** The broad all-dimensional conjecture was disproved in July 2026. Its original two-variable case remains open.

### What 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.

## Status

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](#reference-2) [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

- 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 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
- The formulation and changed status were checked against the cited reference on 2026-07-22.
- 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: Every invertible affine-linear plane map satisfies the condition and has a polynomial inverse.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Computational notes

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

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `plane-jacobian-conjecture`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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 https://mathworld.wolfram.com/JacobianConjecture.html
   - Also cited at Problem statement and variants
   - Also cited at Editorial research route recorded 2026-07-31
   - website; primary source; checked 2026-07-31
   - Source 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.
   - 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.
2. <a id="reference-2"></a>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 https://doi.org/10.1090/S0273-0979-1982-15032-7
   - journal_article; primary source; checked 2026-08-01
   - Source use: original_summary
   - Primary source for the cubic-homogeneous reduction.