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A set on which the local Łojasiewicz exponent is attained. (English) Zbl 0912.32027
From the introduction: “In [Ann. Pol. Math. 67, No. 2, 191-197 (1997)] the authors showed that for a polynomial mapping $$F=(f_1, \dots,f_m): \mathbb{C}^n\to \mathbb{C}^m$$, $$n\geq 2$$, the Łojasiewicz exponent $${\mathcal L}_\infty (F)$$ of $$F$$ at infinity is attained on the set $$\{z\in \mathbb{C}^n: f_1(z) \cdot \dots \cdot f_m (z) =0\}$$. The purpose of this paper is to prove an analogous result for the Łojasiewicz exponent $${\mathcal L}_0(F)$$, where $$F:U\to \mathbb{C}^m$$ is a holomorphic mapping, $$F(0)=0$$ and $$U$$ is a neighbourhood of $$0\in \mathbb{C}^n$$ (Theorem 1). From this result we easily obtain a strict formula for $${\mathcal L}_0 (F)$$ in the case $$n=2$$ and $$m\geq 2$$ in terms of multiplicities of some mappings from $$U$$ into $$\mathbb{C}^2$$ defined by components of $$F$$ (Theorem 2). It is a generalization of Main Theorem from [the authors, Singularities, Banach Cent. Publ. 20, 139-146 (1988; Zbl 0674.32004)]. The proof of this theorem has been simplified by A. Płoski in Ann. Pol. Math. 51, 275-281 (1990; Zbl 0764.32012). His proof has been an inspiration to write this paper”.

##### MSC:
 32S05 Local complex singularities 32S70 Other operations on complex singularities
##### Keywords:
polynomial mapping; Łojasiewicz exponent
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