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

### Tom 88 / 2006

Annales Polonici Mathematici 88 (2006), 193-204 MSC: 14R15, 14E40. DOI: 10.4064/ap88-3-1

#### Streszczenie

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.

#### Autorzy

• Nguyen Van ChauHanoi Institute of Mathematics
18 Hoang Quoc Viet
Hanoi,Vietnam
e-mail
• Carlos GutierrezDepartamento de Matemática
ICMC–USP
Caixa Postal 668
13560–970, São Carlos, SP, Brazil
e-mail

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