On nonsingular polynomial maps of ℝ²

Nguyen Van Chau; Carlos Gutierrez

Annales Polonici Mathematici, Tome 89 (2006), p. 193-204 / Harvested from The Polish Digital Mathematics Library

Résumé

We consider nonsingular polynomial maps F = (P,Q): ℝ² → ℝ² under the following regularity condition at infinity $(J_{\infty})$ : There does not exist a sequence $(p_{k}, q_{k}) \subset ℂ ²$ of complex singular points of F such that the imaginary parts $(ℑ (p_{k}), ℑ (q_{k}))$ tend to (0,0), the real parts $(ℜ (p_{k}), ℜ (q_{k}))$ tend to ∞ and $F (ℜ (p_{k}), ℜ (q_{k}))) \to a \in ℝ ²$ . It is shown that F is a global diffeomorphism of ℝ² 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 |c| large enough.

Publié le : 2006-01-01

Zbl 1102.14043

EUDML-ID : urn:eudml:doc:280387

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     title = {On nonsingular polynomial maps of $\mathbb{R}$$^2$},
     journal = {Annales Polonici Mathematici},
     volume = {89},
     year = {2006},
     pages = {193-204},
     zbl = {1102.14043},
     language = {en},
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Nguyen Van Chau; Carlos Gutierrez. On nonsingular polynomial maps of ℝ². Annales Polonici Mathematici, Tome 89 (2006) pp. 193-204. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap88-3-1/