nLab Jacobian conjecture

Statement

The Jacobian conjecture asserts that: In the category of smooth complex algebraic varieties with regular maps between them, any étale map $F \colon \mathbb{C}^n \to \mathbb{C}^n$ is an isomorphism.

In simpler terms, suppose we are given a map $F = (F_1, \dots, F_n) \colon \mathbb{C}^n \to \mathbb{C}^n$ such that the $F_i$ are polynomials. Then the Jacobian conjecture says that if the differential of $F$ is invertible at each point, $F$ itself is invertible.

State of the research

The conjecture was originally stated by Kraus 1884, Weyr 1886 for $n = 2$, and later stated for $n$ dimensions by Keller 1939. It is known to hold at least for those maps $F$ which have a rational inverse. There were many failed attempts to prove the conjecture, especially in $n = 2$.

An explicit counterexample for $n = 3$ was stated by Alpöge 2026: the polynomial map $F \colon \mathbb{C}^3 \to \mathbb{C}^3$ given by

$$F(x,y,z) \;=\; \big( y^2 (3x y+4)(x y+1) + z(x y + 1)^3, 3x y^2(3x y + 4) + 3x z(x y+1)^2 + y, 2x - x^3 z - 3x^2 y \big)$$

has $det \partial F_i / \partial x_j = -2$ everywhere but $F$ is not one-to-one since for example

$$F(0, 0, -1/4) \;=\; F(1, -3/2, 13/2) \;=\; F(-1, 3/2, 13/2) \;=\; (-1/4, 0, 0) \mathrlap{\,.}$$

This counterexample for $n = 3$ implies counterexamples for all $n \geq 3$ (Zhang 2026).

The conjecture remains open for $n = 2$ (the case for which it was originally stated by Kraus 1884, Weyr 1886).

Literature

General

The conjecture was originally stated for two variables in:

• Ludwig Kraus: Über Functionaldeterminanten, Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften in Wien, Mathematisch-Naturwissenschaftliche Klasse. 90 (1884) 813–826.

• Eduard Weyr: Život a působení dra Ludvíka Krause, Časopis pro pěstování mathematiky a fysiky 15 2 (1886) 49–-52 [dml:122228, pdf]

and later stated for $n$ variables in:

• Ott-Heinrich Keller: Ganze Cremona-Transformationen, Monatshefte für Mathematik und Physik 47 (1939) 299–306 [doi:10.1007/BF01695502]

Surveys:

• Hyman Bass, E. H. Connell, D. Wright: The Jacobian conjecture: Reduction of degree and formal expansion of the inverse, Bull Amer. Math. Soc. 7 (1982)

• Wikipedia: Jacobian conjecture

• A. van den Essen: Jacobian conjecture, Springer Encyclopedia of Mathematics [eom:J/j120010]

See also:

• Arno van den Essen: Polynomial automorphisms and the Jacobian conjecture, Algèbre non commutative, groupes quantiques et invariants (Reims, 1995) 55–81, Sémin. Congr., 2, Soc. Math. France, Paris (1997) [pdf]

• Arno van den Essen: Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics 190 Birkhäuser Verlag (2000) [ISBN:3-7643-6350-9]

The Jacobian conjecture is equivalent to the Dixmier conjecture: every endomorphism of the $r$-th Weyl algebra $A_{r,k}$ over $k$ is an automorphism for all $r$. This is a statement of:

• Alexei Belov-Kanel, Maxim Kontsevich: The Jacobian conjecture is stably equivalent to the Dixmier conjecture, Mosc. Math. J. 7 2 (2007) 209–218 [math.RA/0512171]

which does contain an error in the proof, that has been later amended by others. A shorter algebraic proof is given in

• V. Bavula: The $\text{Jacobian Conjecture}_{2n}$ implies the $\text{Dixmier Problem}_n$ [math.AG/0512250]

It is known that each endomorphism of the $r$-th Weyl algebra is injective, and the Jacobian conjecture claimed it must also be surjective. The disproof of the Jacobian conjecture gives an explicit counterexample.

A related statement by Maxim Kontsevich on automorphisms of Weyl algebras is here:

• Alexei Kanel-Belov, Andrey Elishev, Jie-Tai Yu, Automorphisms of Weyl Algebra and a Conjecture of Kontsevich, arxiv/1802.01225

There is an interesting blog discussion, from the point of view of algebraic geometry:

• David Speyer at secret blogging seminar: how not to prove the Jacobian conjecture

On the origin of the Jacobian conjecture:

• Lázaro Orlando Rodríguez Díaz (June 5, 2026). On the origin of the Jacobian conjecture. Comptes Rendus. Mathématique. 364 (G2): 363–370 [arXiv:2512.23614, doi:10.5802/crmath.831, ISSN:1778-3569]

Counterexamples

A counterexample for $n = 3$ was announced by:

• Levent Alpöge (20 July 2026) [x.com:2079028340955197566]

The observation that this implies counterexamples for all $n \geq 3$:

• Zihan Zhang: Direct Consequences of the Three-Dimensional Counterexample to the Jacobian Conjecture (20 July 2026)

A coordinate-free explanation of the three-dimensional counterexample appears in:

• Remy van Dobben de Bruyn, Divisors in projective bundles over the projective line whose complement is affine space [arXiv:2608.27341]

category: algebra