On nonsingular polynomial maps of $\mathbb R^2$
Volume 88 / 2006
Annales Polonici Mathematici 88 (2006), 193-204
MSC: 14R15, 14E40.
DOI: 10.4064/ap88-3-1
Abstract
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.
