The variety of the asymptotic values of a real polynomial etale map Academic Article uri icon

abstract

  • A polynomial map F : R 2 → R 2 is said to satisfy the Jacobian condition if ∀( X , Y ) ϵ R 2 , 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 : 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 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 , 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 asymptotic 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 ) ϵ R [ X , Y ], which ‘capture’ the whole set of asymptotic values of F .

publication date

  • January 1, 1996