Math Conjectures

Plane Jacobian Conjecture

OPENLandmarkCanonical finite caseProposed 1939 · Canonical special case

Canonical statement

If F=(F1,F2):C2→C2F=(F_1,F_2):\mathbb C^2\to\mathbb C^2 is a polynomial map and
det⁡ ⁣(∂Fi∂xj)1≤i,j≤2∈C×, \det\!\left(\frac{\partial F_i}{\partial x_j}\right)_{1\le i,j\le 2} \in\mathbb C^\times,
then there is a polynomial map G:C2→C2G:\mathbb C^2\to\mathbb C^2 satisfying G∘F=F∘G=id⁡C2G\circ F=F\circ G=\operatorname{id}_{\mathbb C^2}.
View source LaTeX
If \(F=(F_1,F_2):\mathbb C^2\to\mathbb C^2\) is a polynomial map and
\[
  \det\!\left(\frac{\partial F_i}{\partial x_j}\right)_{1\le i,j\le 2}
    \in\mathbb C^\times,
\] then there is a polynomial map \(G:\mathbb C^2\to\mathbb C^2\) satisfying \(G\circ F=F\circ G=\operatorname{id}_{\mathbb C^2}\).
ALG-POLY-001:[Keller1939][BassConnellWright1982][Druzkowski1983][JacobianCounterexampleAFP2026]
Algebra and representation theoryAlgebraic geometry

Notes

The plane Jacobian conjecture asks whether every polynomial map F:C2→C2F:\mathbb C^2\to\mathbb C^2 whose Jacobian determinant is a nonzero constant must be invertible, with a polynomial inverse. The question goes back to Keller's 1939 paper on entire Cremona transformations [Keller1939], and in dimension one the analogous statement is elementary.

For decades the problem was studied in all dimensions at once. Bass, Connell and Wright reduced the general conjecture to maps of degree three [BassConnellWright1982], and Drużkowski developed an effective approach with further reductions [Druzkowski1983]. This landscape changed in July 2026, when an explicit counterexample in dimension three, formally verified in both Isabelle/HOL and Lean, refuted the general conjecture [JacobianCounterexampleAFP2026]; padding with identity coordinates then refutes it in every dimension n≥3n\ge 3.

The counterexample makes no claim about dimension two, which is now the only unresolved case, and the classical reductions pass through higher-dimensional maps, so they no longer bear on it directly. The plane case JC2\mathrm{JC}_2 therefore stands sharply isolated and remains open: settling it requires either a genuinely two-dimensional proof of invertibility or a plane counterexample.

Proof-claim watch (1)

Full Jacobian conjecture

Grade A

Verified refutation for n >= 3; the plane case remains open.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.