The Jacobian conjecture in the plane, first stated by Keller (1939), 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 automorphismiff
the 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).
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.
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of ![C[x,y]](/images/equations/JacobianConjecture/Inline2.svg)
(the polynomial ring in two variables over the complex numbers 
) to itself that fixes 
and sends 
, 
to 
, 
respectively, 
is an automorphism iff
the 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).
