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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”.

32S05 Local complex singularities
32S70 Other operations on complex singularities
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