nLab Jacobian conjecture

Statement

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.

State of the research

The conjecture has been stated by Keller in 1939 and is still open. 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.

Literature

A survey:

• 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)

• 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

• 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, which has been later amended by others. It is 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

category: algebra