The variety of the asymptotic values of a real polynomial etale map Original Research Article

A polynomial map F: R2 → R2 is said to satisfy the Jacobian condition if for all(X, Y)ε R2, J(F)(X, Y) ≠ 0. The real Jacobian conjecture was the assertion that such a map is a global diffeomorphism. Recently the conjecture was shown to be false by S. Pinchuk. According to a theorem of J. Hadamard any counterexample to the conjecture must have asymptotic values. We give the structure of the variety of all the asymptotic values of a polynomial map F: R2 → R2 that satisfies the Jacobian condition. We prove that the study of the asymptotic values of such maps can be reduced to those maps that have only X- or Y-finite asymptotic values. We prove that a Y-finite asymptotic value can be realized by F along a rational curve of the type (X− k, A0 + A1 X + … + AN − 1 XN − 1 + YXN), where X → 0, Y is fixed and K, N> 0 are integers. More precisely we prove that the coordinate polynomials P(U, V) of F(U, V) satisfy finitely many asymptotic identities, namely, identities of the following type, P(X− k, A0 + A1 X + … + AN − 1 XN − 1 + YXN) = A(X, Y)ε R[X, Y], which ‘capture’ the whole set of asymptotic values of F.