The 2-$d$ Jacobian conjecture, the $d$-inversion approximation and its natural boundary

R. Peretz

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XVII, Tome 373 (2009), pp. 210-225

Résumé

Let $F\in\mathbb C[X,Y]^2$ be an étale mapping of degree $\operatorname{deg}F=d$. An Étale mapping $G\in\mathbb C[X,Y]^2$ is called a $d$-inverse approximation of $F$ if $\operatorname{deg}G\le d$ and $F\circ G=(X+A(X,Y),Y+B(X,Y))$ and $G\circ F=(X+C(X,Y),Y+D(X,Y))$ where the orders of the four polynomials $A,B,C$ and $D$ are greater that $d$. 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 2-dimensional Jacobian Conjecture, then $F$ has no $d$-inverse approximation. We also give few conclusions of this result. Bibl. – 18 titles.