Dynamics, points and places at infinity, and the inversion of polynomial self-maps of [formula omitted

Abstract

In this partly expository paper we give two applications of ideas from dynamical systems to the study of the injectivity properties of a polynomial local diffeomorphism F=(F1,F2):R2→R2 (by the work of Pinchuk, these maps need not be globally injective). I) The Jacobian conjecture claims that all polynomial local biholomorphisms G=(G1,G2):ℂ2→ℂ2 must be injective. By the Abhyankar–Moh theory in algebraic geometry, G is injective if G1 has one place at infinity. We prove that this result carries over to the real case, in the following form. It is shown that F is injective if the (possibly singular) complex curve C of {F1=0} is irreducible, its projectivization C˜ has only one point at infinity, and the said point is covered only once by a desingularization ℛ→C˜. II) In our second application we show that every polynomial local diffeomorphism of R2 into itself must be injective at least on certain large regions that contain sequences of disjoint discs of arbitrarily large radii.