Hokkaido Mathematical Journal

VUI Ha Huy, DUC Nguyen Hong,

A formula for the \L ojasiewicz exponent at infinity in the real plane via real approximations.

Hokkaido Mathematical Journal, 38 (2009) pp.417-425

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Abstract

We compute the \L ojasiewicz exponent of $f=(f_1,\ldots,f_n)\colon \Bbb R^2\to\Bbb R^n$ via the real approximation of Puiseux's expansions at infinity of the curve $f_1\ldots f_n=0$. As a consequence we construct a collection of real meromorphic curves which provide a testing set for properness of $f$ as well as a condition, which is very easy to check, for a local diffeomorphism to be a global one.

MSC(Primary)14R25
MSC(Secondary)32A20, 32S05
Uncontrolled Keywords\L ojasiewicz exponent at infinity, Puiseux expansion at infinity, Testing sets for properness of polynomial mappings.