THE VARIETY OF THE ASYMPTOTIC VALUES OF A REAL POLYNOMIAL ETALE MAP

A polynomial map F:R(2) --> R(2) is Said to satisfy the Jacobian condi tion if For All(X, Y) is an element of R(2), J(F)(X, Y) not equal 0. T he real Jacobian conjecture was the assertion that such a map is a glo bal 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:R(2) -- > R(2) that satisfies the Jacobian condition. We prove that the study of the asymptotic values of such maps can be reduced to those maps tha t 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 t ype (X(-k), A(0) + A(1)X + ... A(N-1)X(N-1) + YX(N)), 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 asympt otic identities, namely, identities of the following type, P(X(-k), A( 0) + A(1)X + ... + A(N-1)X(N-1) + YX(N)) = A(X, Y) is an element of R[ X, Y], which 'capture' the whole set of asymptotic values of F.