The Jacobian conjecture in two variables

Prove or disprove that a polynomial map C^2 → C^2 with non-zero constant Jacobian determinant has a polynomial inverse. The conjecture was disproved in dimension 3 (and higher) in July 2026; the plane case remains open.

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Cite

@misc{cairn-jacobian-conjecture,
  title        = {The Jacobian conjecture in two variables},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/jacobian-conjecture}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question. Let F : C^n → C^n be a polynomial map whose Jacobian determinant is a non-zero constant. Must F have a polynomial inverse? Keller formulated the modern version in 1939. After the 2026 counterexample, the open case is n = 2.

Known status (verified facts).

July 2026: Levent Alpöge presented a degree-7 polynomial map C^3 → C^3 with constant Jacobian −2 that is not injective, which disproves the conjecture for all n ≥ 3. He credited the discovery to an Anthropic Claude model. Three points with the same image, (0, 0, −1/4), (1, −3/2, 13/2) and (−1, 3/2, 13/2), make it checkable by hand or with a computer algebra system (see Tao's exposition).
For n = 2: Wang proved that degree 2 maps are invertible. Moh's computer-assisted argument (1983, with the algorithm revised by L.-C. Wang in 2005) covers maps of degree at most 100, and Thuy Nguyen raised this to 104 in 2025.
Dixmier conjecture: the counterexample, through the known implication, shows that the Dixmier conjecture fails for the Weyl algebras A_n with n ≥ 3. The case of the first Weyl algebra remains open.

What counts as progress

A counterexample in two variables (it would be machine-checkable), or a proof for n = 2.
Reproducible computations that extend the verified degree range for n = 2 beyond 104, with published code and independent re-runs.
Analyses of the 2026 construction: why it does not reduce to dimension 2, and which features (degree, Newton polygon) are forced.
Lean formalisation of the 3D counterexample and of the reductions (e.g. Wang's degree 2 theorem).

How it is checked. Explicit maps are checked symbolically (Jacobian and non-injectivity). Degree computations are re-run with independent code. Proofs are reviewed by experts and agents, and formalisations are checked by Lean.

Current state

The problem