nLab Jacobian conjecture

Jacobian conjecture: Let $k$ be an algebraically closed field of characteristics zero, $n\geq 2$ and $\phi: k^n\to k^n$ a (regular) endomorphism of $k^n$ with constant Jacobian (the determinant of the Jacobian matrix, which is in this polynomial case algebraically defined). Then $\phi$ is a regular automorphism, i.e. has a polynomially defined inverse.

The conjecture is open still stated by Keller in 1939. There were many failed attempts to prove the Jacobian conjecture, especially for $n = 2$; there are also some reductions to special cases. For example, it is known that the Jacobian conjecture holds iff it holds for $\phi$ a polynomial map of degree 3. The Jacobian conjecture is also known to hold at least for those $\phi$ which have a rational inverse.

• A. van den Essen, Jacobian conjecture, Springer Online Encyclopedia of Mathematics

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

• Arno van den Essen, Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics, 190. Birkhäuser Verlag, Basel, 2000. xviii+329 pp. ISBN: 3-7643-6350-9

The Jacobian conjecture is also 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 the article

• A. Belov-Kanel, M. 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, which has been later amended by others. It is actually known that each endomorphism of the $r$-th Weyl algebra is injective, and not known wheather it is surjective. A shorter algebraic proof is given in

• V. Bavula, The ${Jacobian Conjecture}_{2n}$ implies the ${Dixmier Problem}_n$, math.AG/0512250

There is a recent proof of related Kontsevich’s statement on automorphisms of Weyl algebra

• 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