The two-dimensional Jacobian conjecture: invertibility of planar Keller maps

The two-dimensional Jacobian conjecture asks whether every polynomial map $(P, Q)\colon \mathbb{C}^2 \to \mathbb{C}^2$ whose Jacobian determinant is a nonzero constant admits a polynomial inverse. It is the surviving open case of Keller's 1939 problem: Liu's preprint (arXiv:2002.10249) is an unrefereed claimed proof of the entire Jacobian conjecture whose key properness claim for Drużkowski maps was refuted by Truong in 2020, and the underlying universal statement was refuted in every dimension $n \ge 3$ in July 2026 (an explicit three-dimensional counterexample, its infinite family, its geometric explanation, and its generalization to all higher dimensions). Any planar counterexample must have coordinate degree at least 108, and this is the only dimension of Keller's 1939 problem that remains undecided.

Background

A polynomial map is a vector $F = (P_1, \dots, P_n)$ of polynomials in $n$ variables, regarded as a self-map $F\colon \mathbb{C}^n \to \mathbb{C}^n$; its Jacobian matrix $JF$ is the matrix of first partial derivatives, and $\det JF$ is again a polynomial. If $F$ has a polynomial inverse (such an $F$ is called an automorphism of affine space), the chain rule forces $\det JF$ to be a nonzero constant. Keller, in a 1939 study of polynomial changes of coordinates (entire Cremona transformations), asked for the converse: if $\det JF$ is a nonzero constant — such an $F$ is called a Keller map — must $F$ be an automorphism? By the inverse function theorem a Keller map is everywhere locally invertible (unramified, in algebraic-geometric language étale), so the question is whether local invertibility forces global invertibility. The affirmative assertion for all $n \ge 2$ became the Jacobian conjecture; by the Lefschetz principle (transfer of first-order algebraic statements between characteristic-zero fields) it is equivalent over any field of characteristic zero, it fails in positive characteristic, and the one-variable case is elementary. Since an injective polynomial self-map of $\mathbb{C}^n$ is automatically an automorphism (a classical fact in the theory of polynomial maps), the conjecture is equally a question about injectivity.

Known counterexamples and reductions delimit the problem sharply. Over the real numbers the strong real Jacobian conjecture is false: Pinchuk (1994) constructed a polynomial local diffeomorphism $\mathbb{R}^2 \to \mathbb{R}^2$ with a nowhere-vanishing (everywhere positive, non-constant) Jacobian determinant that is not injective, so the complex setting is essential and the constant-determinant real case remains open. In the positive direction, Yagzhev (1980) showed that the full conjecture is equivalent to its restriction to maps of the form $x + H(x)$ with $H$ a vector of homogeneous cubic polynomials (Yagzhev maps), and Drużkowski (1983) showed that it is equivalent to its restriction to the cubic-linear maps $F_A(x) = x + (Ax)^{*3}$, where $*$ denotes the componentwise (Hadamard) power and $A$ is a square matrix (Drużkowski maps). Because a local diffeomorphism of $\mathbb{C}^n$ is a global diffeomorphism exactly when it is proper — preimages of compact sets are compact (Hadamard's theorem) — proving properness of these cubic classes became the standard attack line. Liu (2020) claimed a proof of the full conjecture by an optimization argument for the properness of Drużkowski maps, and Truong (2020) refuted that proof by exhibiting matrices $A$ for which Liu's argument would force properness of $F_A$ even though $F_A$ is not proper.

In July 2026 the general conjecture fell. Alpöge announced an explicit three-dimensional Keller map that is not injective; Gallagher produced an infinite family of such maps in dimension three realizing every generic fiber degree (the number of preimages of a general point) $n \ge 3$; Speyer identified the underlying geometry — a tangent sweep of a plane curve, which classical projective duality forces to hit a general point several times while the Jacobian determinant stays identically $1$; and Gao generalized the construction to every dimension $n > 2$, verifying all identities in exact rational arithmetic and computing fiber structures with Gröbner bases (standard canonical generating sets for polynomial ideals). The Jacobian conjecture is therefore false in every dimension $n \ge 3$, and the two-dimensional case — Keller's original question — is the only dimension left undecided. In the plane the problem is usually phrased through Jacobian pairs $(P, Q)$: pairs of complex polynomials in two variables whose Jacobian determinant $\partial_x P\, \partial_y Q - \partial_y P\, \partial_x Q$ is a nonzero constant. The sharpest quantitative constraint known is that any counterexample must satisfy $\max(\deg P, \deg Q) \ge 108$, with all degree pairs below 125 excluded except $(72, 108)$ and $(108, 72)$ (Guccione, Guccione, Horruitiner and Valqui, 2022). The question stated below is the surviving planar case of Keller's problem, item 16 of Smale's 1998 list of mathematical problems for the next century.

Problem Statement

Is every planar Keller map an automorphism? Precisely: if $P, Q \in \mathbb{C}[x, y]$ are polynomials whose Jacobian determinant satisfies

$$\frac{\partial P}{\partial x}\,\frac{\partial Q}{\partial y} - \frac{\partial P}{\partial y}\,\frac{\partial Q}{\partial x} = c \quad\text{for some } c \in \mathbb{C}^{*},$$

must the map $(P, Q)\colon \mathbb{C}^2 \to \mathbb{C}^2$ admit a polynomial inverse — equivalently, by the injectivity theorem for polynomial self-maps of affine space, must it be injective? The statement may equivalently be posed over any field of characteristic zero. The implication is asserted in full generality, with no degree bound, genericity hypothesis, or auxiliary restriction; a refutation would be an explicit non-injective Jacobian pair $(P, Q)$. This is the two-dimensional case of Keller's 1939 problem — the strong real Jacobian conjecture being already false by Pinchuk's counterexample and the cases $n \ge 3$ having been refuted in July 2026 — and it is the only dimension in which the Jacobian conjecture remains undecided.

References

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