The Jacobian conjecture, the d-inversion approximation and its natural boundary

Let F ∈ ℂ[X,Y]² be an étale map of degree deg F = d. An étale map G ∈ ℂ[X,Y]² is called a d-inverse approximation of F if deg G ≤ d and F ○ G =(X + A(X, Y), Y + B(X, Y)) and G ○ F =(X + C(X, Y), Y + D(X, Y)), where the orders of the four polynomials A, B, C, and D are greater than d. It is a well-known result that every ℂ-automorphism F of degree d has a d-inverse approximation, namely, F^-1. In this paper, we prove that if F is a counterexample of degree d to the two-dimensional Jacobian conjecture, then F has no d-inverse approximation. We also give few consequences of this result. Bibliography: 18 titles.