On nonsingular polynomial maps of $\mathbb R^2$

Nguyen Van Chau ¹   ; Carlos Gutierrez ²

¹ Hanoi Institute of Mathematics 18 Hoang Quoc Viet Hanoi,Vietnam
² Departamento de Matemática ICMC–USP Caixa Postal 668 13560–970, São Carlos, SP, Brazil

Annales Polonici Mathematici, Tome 88 (2006) no. 3, pp. 193-204

Résumé

We consider nonsingular polynomial maps $F = (P,Q):\mathbb R^2 \to \mathbb R^2$ under the following regularity condition at infinity $(J_\infty)$: There does not exist a sequence $\{(p_k,q_k)\}\subset \mathbb C^2$ of complex singular points of $F$ such that the imaginary parts $(\Im (p_k),\Im(q_k))$ tend to $(0,0)$, the real parts $(\Re(p_k), \Re(q_k))$ tend to $\infty$ and $F(\Re(p_k),\Re(q_k)) )\rightarrow a\in \mathbb R^2$. It is shown that $F$ is a global diffeomorphism of $\mathbb R^2$ if it satisfies Condition $(J_\infty)$ and if, in addition, the restriction of $F$ to every real level set $P^{-1}(c) $ is proper for values of $\vert c\vert$ large enough.

Détail

DOI : 10.4064/ap88-3-1

Keywords: consider nonsingular polynomial maps mathbb mathbb under following regularity condition infinity infty there does exist sequence k subset mathbb complex singular points imaginary parts tend real parts tend infty rightarrow mathbb shown global diffeomorphism mathbb satisfies condition infty addition restriction every real level set proper values vert vert large enough