Proof of Two Dimensional Jacobian Conjecture

is a nonzero constant, where A = ( ∂ fi ∂ xj )i,j=1 is the n × n Jacobian matrix of f1, ..., fn. One of the major unsolved problems of mathematics [S] (see also [B, CM, V2]), viz. the Jacobian conjecture, states that the reverse of the above statement also holds, namely, if the Jacobian determinant J(f1, ..., fn) ∈ F , then f1(x1, ..., xn), ..., fn(x1, ..., xn) ∈ F[x1, ..., xn] are generators of F[x1, ..., xn]. For convenience, if (1.1) holds, we shall refer f1, ..., fn to as polynomials with nonzero Jacobian determinant property (or simply, NJDP ). This conjecture relates to many aspects of mathematics [A, ES, H, R, SW, SY] and has attracted great attention in mathematics and physics literature during the past 60 years and there have been a various ways of approaches toward the proof or disproof of this conjecture (here we simply give a short random list of references [BCW, CCS, D, J, K, Ki, KM, V1, V2, W]). Hundreds of papers have appeared in connection with this conjecture, even for the simplest case n = 2 [AO, N, No]. However this conjecture remains unsolved even for the case n = 2. The difficulty in solving this conjecture probably lies in that although the NJDP may contain much information, one is unable to use it. In this self-contained note, we give a proof of the Jacobian conjecture for the case n = 2. The main result is the following.