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A set on which the Łojasiewicz exponent at infinity is attained. (English) Zbl 0924.32004
From the introduction. The purpose of this paper is to prove that the Łojasiewicz exponent at infinity of a polynomial mapping $$F:\mathbb{C}^n\to\mathbb{C}^m$$ is attained on a proper algebraic subset of $$\mathbb{C}^n$$ defined by the components of $$F$$.
As a corollary we obtain a result of Z. Jelonek [“Testing sets for properness of polynomial mappings” (preprint 16, Inst. Math., Jagiellonian Univ. 1996)] on testing sets for properness of polynomial mappings (corollary 3) and a formula for the Łojasiewicz exponent at infinity of $$F$$ in the case $$n=2$$, $$m\geq 2$$, in terms of parametrizations of branches (at infinity) of zeroes of the components of $$F$$ (theorem 2). This result is a generalization of the authors’ result for $$n=m=2$$ [J. Chądzyński and T. Krasiński in: Singularities, Banach Cent. Publ. 20, 147-160 (1988; Zbl 0661.32008), main theorem].

##### MSC:
 32B05 Analytic algebras and generalizations, preparation theorems 14E05 Rational and birational maps 14P05 Real algebraic sets
##### Keywords:
Łojasiewicz exponent at infinity; polynomial mapping
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