On Jacobian conjecture

Any endomorphism of Cn defined by n polynomials with everywhere non-vanishing Jacobian is an automorphism. The Jacobian conjecture originated fromKeller ([5]). Let F1, . . . , Fn ∈ C[x1, . . . , xn] be a set of n polynomials in n variables with n ≥ 1 such that the Jacobian of these polynomials is a nonzero constant. The Jacobian conjecture says that the subalgebra C[F1, . . . , Fn] of C[x1, . . . , xn] is actually equal to C[x1, . . . , xn]. For the history and various approaches to the problem see [1],[7] and [8]. We fix once for all an algebraically closed field k of characteristic zero. Let F : Ak → A n k be a map defined by (x1, . . . , xn) 7→ (F1(x1, . . . , xn), . . . , Fn(x1, . . . , xn)) with every Fi ∈ k[x1, . . . , xn]. Let JF denote the Jacobian matrix (∂Fi/∂xj)1≤i,j≤n. The map F is called a Jacobian map if det(JF ) ∈ k. Theorem Every Jacobian map is an isomorphism. Supported by the National Science Foundation of the People’s Republic of China. 2000 Mathematical Subject Classification : Primary 14R15, Secondary 14E22.