zbMATH — the first resource for mathematics

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).

14H37 Automorphisms of curves
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
Full Text: DOI