# JC₂: the Jacobian conjecture in the plane

- kind: conjecture
- status: open
- year: 1884
- authors: L. Kraus (1884, with a flawed proof); O.-H. Keller (1939, in all dimensions)
- in one line: JC₂ — the one dimension of the Jacobian conjecture still open
- formal statement: `jacobian_conjecture_two_variables, formal-conjectures at 6fbb54f` (https://github.com/google-deepmind/formal-conjectures/blob/6fbb54f24ccc2e64dcfaffc28c58950e377110d2/FormalConjectures/Wikipedia/JacobianConjecture.lean)

**Statement.** If $f,g\in k[x,y]$ over a field of characteristic $0$ satisfy $\partial_xf\,\partial_yg-\partial_yf\,\partial_xg\in k^{\times}$, then $k[f,g]=k[x,y]$.

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](/wiki/Statement:jc-n.md): 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.

**What is left of JC₂** — x: max(deg f, deg g) (1–1000), y: geometric degree (1–16).

- excluded: x <= 104 — degree ≤ 100: Moh 1983; ≤ 104: Nguyen 2025 (Statement:moh-degree-100)
- excluded: y = 1 — birational: Keller 1939 (Statement:keller-birational)
- excluded: y = 2 — Galois: Campbell 1973 (Statement:campbell-galois)
- excluded: y <= 5 — degree ≤ 5: Żołądek 2008 (Statement:plane-gcd-bounds)
- excluded: y is prime — prime: Moskowicz 2024, unrefereed (Statement:prime-degree-plane)
- excluded for the torus class of constructions: no torus-symmetric counterexample: Shaska 2026, unrefereed (Statement:graded-equivariant-keller)

The light cells to the right of the degree band are where a plane counterexample could still be. Their hatching says it could not have a torus symmetry. No point is plotted, because none is known.

A third invariant, the gcd of the component degrees, is walled too: a counterexample needs gcd(deg f, deg g) outside {1,…,16} and not a prime or twice a prime, so at least 18.

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](/wiki/Statement:moh-degree-100.md). 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](/wiki/Statement:graded-equivariant-keller.md): 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](/wiki/Statement:graded-equivariant-keller.md) 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](/wiki/Statement:equivariance-forced-question.md), 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](/wiki/Statement:jc-n.md)). By Tsuchimoto and Belov-Kanel–Kontsevich, [JC₂ implies Dixmier's original problem DC₁ for the first Weyl algebra](/wiki/Statement:dixmier-a1.md).[5] In the other direction, [DC₂](/wiki/Statement:dixmier-a2.md) implies JC₂, and so does the four-variable Hessian conjecture [HC₄](/wiki/Statement:hessian-hc4.md).[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](/wiki/Statement:poisson-conjecture.md)).

Two classical structure theorems frame any attack. By [Jung–van der Kulk](/wiki/Statement:jung-van-der-kulk.md) 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](/wiki/Statement:abhyankar-points-at-infinity.md) 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](/wiki/Statement:pinchuk-real.md) 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](/wiki/Statement:real-keller.md).

## Relations

- special case of [jc-n](/wiki/Statement:jc-n.md) — 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. 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. [1, 2, 3]
- implies [dixmier-a1](/wiki/Statement:dixmier-a1.md) — By Tsuchimoto and Belov-Kanel–Kontsevich, JC₂ implies Dixmier's original problem DC₁ for the first Weyl algebra. In the other direction, DC₂ implies JC₂, and so does the four-variable Hessian conjecture HC₄. The rank-one Poisson conjecture is JC₂ restated: a Poisson endomorphism of $k[x,y]$ is exactly a pair with Jacobian 1 (see PCₙ). [5, 3]

## 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.
2. T. Tao, [A digestion of the Jacobian conjecture counterexample](https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/), 21 July 2026: “The conjecture remains open in two dimensions.”
3. G. Meng, L. Yang, [A five-variable counterexample to the Hessian conjecture, and the low-dimensional status of the Jacobian and Hessian conjectures](https://arxiv.org/abs/2607.22198), arXiv:2607.22198 (24 and 27 July 2026).
4. D. Fong and Claude Fable 5, [claims ledger](https://jacobianconjectures.com/CLAIMS.md), jacobianconjectures.com, from 20 July 2026.
5. A. Belov-Kanel, M. Kontsevich, [The Jacobian conjecture is stably equivalent to the Dixmier conjecture](https://arxiv.org/abs/math/0512171), Moscow Math. J. 7 (2007) 209–218.

## Relations to this page

- equivalent to [keller-plane](/wiki/Problem:keller-plane.md) — This atlas problem is the two-variable Jacobian conjecture JC₂, stated over ℂ.
- related to [abhyankar-points-at-infinity](/wiki/Statement:abhyankar-points-at-infinity.md) — So the plane conjecture is equivalent to the claim that no Jacobian pair has a component with exactly two points at infinity.
- implied by [dixmier-a2](/wiki/Statement:dixmier-a2.md) — The Vaserstein–Kac pullback at $n=2$ turns a non-invertible Keller map of the plane into a non-surjective endomorphism of $A_2$, so DC₂ implies JC₂.
- implied by [equivariance-forced-question](/wiki/Statement:equivariance-forced-question.md) — If O5 holds in dimension two, there is no plane counterexample, so a positive answer to O5 would prove JC₂, through Shaska's plane theorem (itself unrefereed).
- generalizes [graded-equivariant-keller](/wiki/Statement:graded-equivariant-keller.md) — So the plane conjecture holds for every Jacobian pair with a nontrivial torus symmetry.
- implied by [hessian-hc4](/wiki/Statement:hessian-hc4.md) — Therefore HC₄ implies the plane Jacobian conjecture JC₂.
- related to [jung-van-der-kulk](/wiki/Statement:jung-van-der-kulk.md) — This is the structure theorem behind nearly every plane-specific approach. Because every automorphism of the plane is tame, the plane Jacobian conjecture says exactly that every Jacobian pair is a composition of affine and triangular maps.
- generalizes [moh-degree-100](/wiki/Statement:moh-degree-100.md) — He then ran a computer search over degree pairs up to 100, which left six possible cases, and eliminated those separately. The result is that the plane conjecture holds for every Jacobian pair of degree at most 100.
- related to [pinchuk-real](/wiki/Statement:pinchuk-real.md) — What they show is that no proof of JC₂ can rest on real local invertibility and topology alone: any proof of the plane conjecture has to use that the determinant is constant, or that the field is ℂ.
- generalizes [plane-gcd-bounds](/wiki/Statement:plane-gcd-bounds.md) — Each result is the plane conjecture restricted to Jacobian pairs whose degree gcd lies in an excluded set, and proves it there.
- generalizes [prime-degree-plane](/wiki/Statement:prime-degree-plane.md) — Moskowicz's preprint proves that the plane conjecture holds for every Keller map whose field extension has prime degree. The proof splits on whether $xy$ lies in $\mathbb C(p,q)$.

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