The geometry of the asymptotics of polynomial maps Academic Article uri icon

abstract

  • The aim of this paper is to develop a theory for the asymptotic behavior of polynomials and of polynomial maps overR and overC and to apply it to the Jacobian conjecture. This theory gives a unified frame for some results on polynomial maps that were not related before. A well known theorem of J. Hadamard gives a necessary and sufficient condition on a local diffeomorphismf: R n →R n to be a global diffeomorphism. In order to show thatf is a global diffeomorphism it suffices to exclude the existence of asymptotic values forf. The real Jacobian conjecture was shown to be false by S. Pinchuk. Our first application is to understand his construction within the general theory of asymptotic values of polynomial maps and prove that there is no such counterexample for the Jacobian conjecture overC. In a second application we reprove a theorem of Jeffrey Lang which gives an equivalent formulation of the Jacobian conjecture in terms of Newton polygons. This generalizes a result of Abhyankar. A third application is another equivalent formulation of the Jacobian conjecture in terms of finiteness of certain polynomial rings withinC[U, V]. The theory has a geometrical aspect: we define and develop the theory of etale exotic surfaces. The simplest such surface corresponds to Pinchuk's construction in the real case. In fact, we prove one more equivalent formulation of the Jacobian conjecture using etale exotic surfaces. We consider polynomial vector fields on etale exotic surfaces and explore their properties in relation to the Jacobian conjecture. In another application we give the structure of the real variety of the asymptotic values of a polynomial mapf: R 2 →R 2 .

publication date

  • January 1, 1998