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On the Łojasiewicz exponent at infinity for polynomial mappings of $$\mathbb{C}^2$$ into $$\mathbb{C}^2$$ and components of polynomial automorphisms of $$\mathbb{C}^2$$. (English) Zbl 0791.14004
Let $$H=(f,g):\mathbb{C}^ 2 \to \mathbb{C}^ 2$$ be a polynomial mapping and $N(H)=\{\nu \in \mathbb{R}:\;\exists A>0,\exists B>0,\forall | z |>B,\qquad A | z |^ \nu \leq | H(z) |\}.$ By the Łoasiewicz exponent at infinity of $$H$$ the authors mean $$\sup N(H)$$ when $$N(H) \neq \emptyset$$ and $$-\infty$$ when $$N(H)=\emptyset$$. In the paper they give exact and effective formulae for the Łojasiewicz exponent at infinity of $$H$$ in all the possible cases. Moreover, they give also a characterization of the component of a polynomial automorphism of $$\mathbb{C}^ 2$$ (in terms of the Łojasiewicz exponent at infinity).

##### MSC:
 14H37 Automorphisms of curves 13F20 Polynomial rings and ideals; rings of integer-valued polynomials
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