The Jacobian conjecture in the plane states that given a ring map of (the polynomial ring in two variables over the complex numbers ) to itself that fixes  and sends , to ,  respectively, is an automorphism iffthe Jacobian  is a nonzero element of . The condition can easily shown to be necessary, but proving sufficiency has been an open problem since Keller (1939).

The Jacobian conjecture is one of Smale's problems.

There have been at least five published incorrect proofs and many incorrect attempts over the years. In November 2004, Hochster (2004) sent an email announcing a new proof by Carolyn Dean. However, this proof unfortunately contained an error as well.