It was first conjectured in 1939 by Ott-Heinrich Keller,[1] and widely publicized by Shreeram Abhyankar, as an example of a difficult question in algebraic geometry that can be understood using little beyond a knowledge of calculus.
The Jacobian conjecture is notorious for the large number of attempted proofs that turned out to contain subtle errors.
There are currently no known compelling reasons for believing the conjecture to be true, and according to van den Essen[2] there are some suspicions that the conjecture is in fact false for large numbers of variables (indeed, there is equally also no compelling evidence to support these suspicions).
The Jacobian conjecture is number 16 in Stephen Smale's 1998 list of Mathematical Problems for the Next Century.
However, Kossivi Adjamagbo [ht] suggested extending the Jacobian conjecture to characteristic p > 0 by adding the hypothesis that p does not divide the degree of the field extension k(X) / k(F).
[4] Hyman Bass, Edwin Connell, and David Wright showed that the general case follows from the special case where the polynomials are of degree 3, or even more specifically, of cubic homogeneous type, meaning of the form F = (X1 + H1, ..., Xn + Hn), where each Hi is either zero or a homogeneous cubic.
The case where k(X) is a Galois extension of k(F) was proved by Andrew Campbell for complex maps[8] and in general by Michael Razar[9] and, independently, by David Wright.
[11][12] Michiel de Bondt and Arno van den Essen[13][14] and Ludwik Drużkowski[15] independently showed that it is enough to prove the Jacobian Conjecture for complex maps of cubic homogeneous type with a symmetric Jacobian matrix, and further showed that the conjecture holds for maps of cubic linear type with a symmetric Jacobian matrix, over any field of characteristic 0.