The Jacobian conjecture (false for $n \ge 3$; open for $n = 2$)

Jacobian conjecture. Let $n \ge 1$ and let $F = (F_1, \dots, F_n) : \mathbb{C}^n \to \mathbb{C}^n$ be a polynomial map whose Jacobian determinant

$\det\left(\frac{\partial F_i}{\partial x_j}\right)$

is a nonzero constant. Then $F$ is bijective, and its inverse is again a polynomial map.

Status: FALSE for every $n \ge 3$, and open for $n = 2$. Ott-Heinrich Keller posed the $n$-variable conjecture in 1939, and it is number $16$ on Smale's 1998 list of problems for the century. On 19 July 2026 Levent Alpöge announced an explicit three-variable counterexample, found with the assistance of Anthropic's Claude Fable 5. Write $F = (P, Q, R) : \mathbb{C}^3 \to \mathbb{C}^3$ with

$P = (1 + xy)^3 z + y^2 (1 + xy)(4 + 3xy), \qquad Q = y + 3x(1 + xy)^2 z + 3xy^2(4 + 3xy), \qquad R = 2x - 3x^2 y - x^3 z.$

The Jacobian determinant of $F$ is the constant $-2$, so the hypothesis holds; but

$F\left(0, 0, -\tfrac{1}{4}\right) = F\left(1, -\tfrac{3}{2}, \tfrac{13}{2}\right) = F\left(-1, \tfrac{3}{2}, \tfrac{13}{2}\right) = \left(-\tfrac{1}{4}, 0, 0\right),$

so $F$ is three-to-one over that point and is not injective. Adjoining identity coordinates turns this into a counterexample in every dimension $n \ge 3$. The coefficients are rational, so the conjecture fails over every field of characteristic zero. The case $n = 1$ is elementary and true. The two-variable case, the plane Jacobian conjecture, is still open; it is older than Keller's general form, having been stated by Ludwig Kraus in 1884.

Characteristic zero remains essential to the surviving question: in characteristic $p$ the map $x \mapsto x - x^{p}$ has Jacobian determinant $1$ and is not injective, so nothing is being conjectured there.

Neither proved nor disproved in this library. Nothing here depends on the conjecture, and this library develops neither the commutative algebra nor the algebraic geometry in which it was attacked. The refutation is a different matter: it is a finite identity in $\mathbb{Q}[x, y, z]$, and both displayed claims above were re-checked by exact rational arithmetic during the audit of this page. What this library does not contain is the theory the question belongs to, which is why the entry stays here rather than becoming a counterexample item with a proof.

What was known before, and what is now known. The converse direction is easy and is the reason the hypothesis is the natural one: if a polynomial map has a polynomial inverse then the chain rule makes the two Jacobian determinants reciprocal polynomials, and the only units of a polynomial ring over a field are the nonzero constants, so each is constant. The deepest structural result in the hard direction was the reduction of Bass, Connell and Wright (1982): the conjecture in all dimensions follows from the special case $F = X + H$ with $H$ homogeneous of degree $3$ and nilpotent Jacobian matrix, so bounding the degree never helped. The conjecture was also known to be stably equivalent to the Dixmier conjecture on endomorphisms of the Weyl algebra (Tsuchimoto 2005; Belov-Kanel and Kontsevich 2007), the Dixmier conjecture for $A_n$ implying the Jacobian conjecture in $n$ variables. That implication now runs backwards as a refutation: the three-variable counterexample shows the Dixmier conjecture is false for $A_n$ for every $n \ge 3$, while it stays open for $A_1$ and $A_2$ exactly because the plane Jacobian conjecture does.

A neighbouring statement that was already FALSE. Weakening "constant nonzero" to merely "nowhere zero" over $\mathbb{R}$ destroys the conclusion, and this was known long before 2026: Pinchuk (1994) constructed a polynomial map $\mathbb{R}^2 \to \mathbb{R}^2$ whose Jacobian determinant is everywhere positive and which is not injective. Pinchuk's example says nothing about the plane Jacobian conjecture, whose hypothesis is the stronger one that the determinant is constant.

Why it matters here. The inverse function theorem is in scope for this library, and it is exactly the local statement: a nonvanishing Jacobian determinant gives a local inverse. The Jacobian conjecture asked for the global polynomial upgrade, and the pair was for decades the cleanest illustration available of how much stronger a global statement is than its local counterpart. It is now an illustration of something else as well, which is why the entry is worth keeping rather than deleting: a question can stand for eighty-seven years, be believed by specialists in the plane case and disbelieved in high dimensions, and then be settled by a counterexample a reader can verify with nothing beyond the definition of a determinant.