closed field of characteristic 0. Let k[X] denote the polynomial ring and k[F] denote the k-subalgebra generated by f1, ..., fn. For a given F, the Jacobian conjecture is true if, and only if, . Keller (1939) proved the birational case, that is, where the two fields k(X) and k(F) are equal. The case where k(X) is a Galois extension of k(F) was proved by Andrew Campbell for complex maps and in general by Michael Razar and, independently, by David Wright. Tzuong-Tsieng Moh checked the conjecture for polynomials of degree at most 100 in two variables. Michiel de Bondt and Arno van den Essen and Ludwik Drużkowski 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. The strong real Jacobian conjecture was that a real polynomial map with a nowhere vanishing Jacobian determinant has a smooth global inverse. That is equivalent to asking whether such a map is topologically a proper map, in which case it is a covering map of a simply connected manifold, hence invertible. Sergey Pinchuk constructed two variable counterexamples of total degree 35 and higher. It is well-known that the Dixmier conjecture implies the Jacobian conjecture. Conversely, it is shown by Yoshifumi Tsuchimoto and independently by Alexei Belov-Kanel and Maxim Kontsevich that the Jacobian conjecture for 2N variables implies the Dixmier conjecture in N dimensions. A self-contained and purely algebraic proof of the last implication is also given by Kossivi Adjamagbo and Arno van den Essen who also proved in the same paper that these two conjectures are equivalent to the Poisson conjecture. See also List of unsolved problems in mathematics References External links Web page of Tzuong-Tsieng Moh on the conjecture Polynomials Algebraic geometry Conjectures Unsolved problems in geometry