# The Jacobian conjecture in the plane

- kind: problem
- status: open
- year: 1884
- in one line: Open: the two-variable case, older than Keller

**Statement.** A polynomial map of $\mathbb{C}^2$ with constant nonzero Jacobian determinant is invertible.

Main article: [jc-plane](/wiki/Statement:jc-plane.md)

This atlas problem is [the two-variable Jacobian conjecture JC₂, stated over ℂ](/wiki/Statement:jc-plane.md). The complex form and the form over any field of characteristic zero are the same statement, by the Lefschetz principle. The atlas curation first called it Keller's problem. Kraus stated the two-variable case in 1884, with a flawed proof, and Keller posed the question in all dimensions in 1939.[1]

The 2026 counterexamples all need three variables. What they leave of the plane case is drawn on [the JC₂ page](/wiki/Statement:jc-plane.md):

- degree above 100 (Moh; 104 refereed; 108 claimed);
- geometric degree 6 or more, and not prime if an unrefereed preprint holds;
- a component-degree gcd of at least 18;
- no torus symmetry.

The plane case now controls Dixmier's original problem for the first Weyl algebra, through JC₂ ⇒ DC₁.

What would settle it: the Lean statement `jacobian_conjecture_two_variables` in google-deepmind/formal-conjectures, which is tagged open.

[Algebraic geometry](/wiki/Space:algebraic-geometry.md) · [Mathematics](/wiki/Continent:mathematics.md)

## Relations

- equivalent to [jc-plane](/wiki/Statement:jc-plane.md) — This atlas problem is the two-variable Jacobian conjecture JC₂, stated over ℂ.

## References

1. L. O. Rodríguez Díaz, [On the origin of the Jacobian conjecture](https://arxiv.org/abs/2512.23614), C. R. Math. 364 (2026) 363–370.

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