Jacobian conjecture in dimension two

Let $K$ be a field of characteristic $0$. For polynomials $p,q\in K[x,y]$, the Jacobian determinant of the map $F=(p,q)\colon K^2\to K^2$ is

$$J_F=\det\begin{pmatrix}\partial p/\partial x & \partial p/\partial y\\ \partial q/\partial x & \partial q/\partial y\end{pmatrix}=p_xq_y-p_yq_x\in K[x,y].$$

Conjecture (plane case). For every field $K$ of characteristic $0$ and all $p,q\in K[x,y]$: if $J_F=c$ for some constant $c\in K\setminus\{0\}$, then $F$ has a polynomial inverse. That is, there are $g,h\in K[x,y]$ with

$$g(p,q)=x,\quad h(p,q)=y,\quad p(g,h)=x,\quad q(g,h)=y .$$

Equivalently, $K[p,q]=K[x,y]$. It suffices to prove this for $K=\mathbb C$ [2]. A disproof is an explicit pair $p,q$ with constant nonzero Jacobian such that $F$ is not invertible, for instance not injective on $\mathbb C^2$.

Keller's original question asked the same for every dimension $n$. That general form is false for every $n\ge3$: in July 2026 an explicit counterexample with rational coefficients was announced for $n=3$,

$$F(z_1,z_2,z_3)=\Bigl((1+z_1z_2)^3z_3+z_2^2(1+z_1z_2)(4+3z_1z_2),\;\;z_2+3z_1(1+z_1z_2)^2z_3+3z_1z_2^2(4+3z_1z_2),\;\;2z_1-3z_1^2z_2-z_1^3z_3\Bigr),$$

with $J_F=-2$ and $F(0,0,-\tfrac14)=F(1,-\tfrac32,\tfrac{13}2)=F(-1,\tfrac32,\tfrac{13}2)$. Adjoining identity coordinates gives counterexamples in all higher dimensions [5, 6, 7]. Only $n=2$ remains open, and it is the problem stated above.

Background

Keller posed the problem in 1939 for maps with integer coefficients [1]. Over $\mathbb C$ a map with constant nonzero Jacobian is a local biholomorphism everywhere, so the question asks whether this local invertibility forces global polynomial invertibility. Białynicki-Birula and Rosenlicht showed that an injective polynomial self-map of $\mathbb C^n$ is bijective with polynomial inverse [3]. So the plane case is equivalent to injectivity of every Keller map of $\mathbb C^2$.

Bass, Connell and Wright's survey [2] fixed the modern formulation and catalogued many published false proofs. The problem was on Smale's list of problems for the twenty-first century. The general conjecture is stably equivalent to the Dixmier conjecture on endomorphisms of Weyl algebras [4]. The counterexamples for $n\ge3$ do not settle the plane case, and they do not settle the Dixmier conjecture for the first Weyl algebra.