nLab Jacobian conjecture

Jacobian conjecture: Let kk be an algebraically closed field of characteristics zero, n2n\geq 2 and ϕ:k nk n\phi: k^n\to k^n a (regular) endomorphism of k nk^n with constant Jacobian (the determinant of the Jacobian matrix, which is in this polynomial case algebraically defined). Then ϕ\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=2n = 2; there are also some reductions to special cases. For example, it is known that the Jacobian conjecture holds iff it holds for ϕ\phi a polynomial map of degree 3. The Jacobian conjecture is also known to hold at least for those ϕ\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 rr-th Weyl algebra A r,kA_{r,k} over kk is an automorphism for all rr. 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 rr-th Weyl algebra is injective, and not known wheather it is surjective. A shorter algebraic proof is given in

  • V. Bavula, The JacobianConjecture 2n{Jacobian Conjecture}_{2n} implies the DixmierProblem n{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:

Last revised on June 8, 2018 at 23:10:14. See the history of this page for a list of all contributions to it.