JC₂: the Jacobian conjecture in the plane

Two polynomials $f,g$ in two variables whose Jacobian determinant is a nonzero constant are called a Jacobian pair. The conjecture says every Jacobian pair generates the whole polynomial ring, so the map $(f,g)$ has a polynomial inverse. The two-variable statement is the oldest form of the problem: Kraus stated it in 1884, with a proof that fails at its last step.^[1] It is the case $n=2$ of the general conjecture, and after July 2026 it is the only dimension of JCₙ that is still open: every counterexample found that month needs at least three variables.^[2][3]

What is left of it

A counterexample in the plane would be a point of this chart. Its coordinates are the larger component degree and the geometric degree, the number of preimages of a general point. Each band is a theorem that no counterexample lives there. The hatching marks a whole mechanism ruled out, rather than a region of the chart.

The walls are of different strengths. Moh's degree bound is refereed but computer-assisted. Wang repaired its algorithm in 2005, and Xu argues that one case, (99, 66), is still incomplete (unrefereed). Guccione, Guccione, Horruitiner and Valqui independently discard every degree pair below 125 except (72, 108), in an unrefereed preprint; see Moh's theorem. The fiber-degree rows 1 and 2 hold in every dimension. Rows 3 to 5 are plane-specific: Orevkov excluded degree 3 in 1986, Domrina degree 4, and Żołądek proved the conjecture for topological degree at most 5 in 2008. The prime rows rest on one unrefereed preprint. The torus hatching is Shaska's theorem: a plane Keller map with any nontrivial $\mathbb G_m$-symmetry is an automorphism. The Fong–Fable ledger recorded a two-line version of the same wall for torus-equivariant twists on 20 July 2026.^[4]

Why the three-dimensional constructions stop here

Alpöge's map, Gallagher's family, the factory maps and Harish's map are all equivariant for a torus action with weights of both signs. Shaska proves that a graded counterexample needs mixed-sign weights and at least three variables, and in the plane every nontrivial symmetry forces an automorphism. So a plane counterexample, if one exists, has no torus symmetry at all, and it needs a different mechanism. Whether counterexamples in higher dimensions must carry such a symmetry is the open question O5, and by Shaska's theorem its plane form would prove JC₂.

What it controls

JC₂ is the hinge of four families of conjectures (see the ladder on JCₙ). By Tsuchimoto and Belov-Kanel–Kontsevich, JC₂ implies Dixmier's original problem DC₁ for the first Weyl algebra.^[5] In the other direction, DC₂ implies JC₂, and so does the four-variable Hessian conjecture HC₄.^[3] The rank-one Poisson conjecture is JC₂ restated: a Poisson endomorphism of $k[x,y]$ is exactly a pair with Jacobian 1 (see PCₙ).

Two classical structure theorems frame any attack. By Jung–van der Kulk every automorphism of the plane is a composition of affine and triangular maps, so JC₂ says every Jacobian pair is one of these. By Abhyankar each component of a Jacobian pair has at most two points at infinity, and a counterexample of minimal degree has exactly two.

Over the reals the constant determinant is essential. Pinchuk's maps of ℝ² have an everywhere positive, non-constant Jacobian and are not injective. JC₂ implies the real plane statement, whose status is recorded on the real Jacobian question.

1. ↑ L. O. Rodríguez Díaz, On the origin of the Jacobian conjecture, C. R. Math. 364 (2026) 363–370.
2. ↑ T. Tao, A digestion of the Jacobian conjecture counterexample, 21 July 2026: “The conjecture remains open in two dimensions.”
3. ↑ a b G. Meng, L. Yang, A five-variable counterexample to the Hessian conjecture, and the low-dimensional status of the Jacobian and Hessian conjectures, arXiv:2607.22198 (24 and 27 July 2026).
4. ↑ D. Fong and Claude Fable 5, claims ledger, jacobianconjectures.com, from 20 July 2026.
5. ↑ A. Belov-Kanel, M. Kontsevich, The Jacobian conjecture is stably equivalent to the Dixmier conjecture, Moscow Math. J. 7 (2007) 209–218.