The variety of the asymptotic values of a real polynomial etale map

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Abstract

A polynomial map F : R2 → R2 is said to satisfy the Jacobian condition if ∀(X, Y) ∈ 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 + A1X + ⋯ + 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 + A1X + ⋯ + AN-1X N-1 + YXN) = A(X, Y) ∈ R[X, Y], which 'capture' the whole set of asymptotic values of F.

Original languageEnglish
Pages (from-to)103-112
Number of pages10
JournalJournal of Pure and Applied Algebra
Volume106
Issue number1
DOIs
StatePublished - 15 Jan 1996

ASJC Scopus subject areas

  • Algebra and Number Theory

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