The Jacobian conjecture, the d -inversion approximation and its natural boundary Academic Article uri icon

abstract

  • Let $$ F \in \mathbb{C}{\left[ {X,\,Y} \right]^2} $$ be an étale map of degree deg F = d. An étale map $$ G \in \mathbb{C}{\left[ {X,Y} \right]^2} $$ 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 $$ {\mathbb{C}^2} $$-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.

publication date

  • July 1, 2010