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The Łojasiewicz exponent of an analytic mapping of two complex variables at an isolated zero. (English) Zbl 0674.32004
Singularities, Banach Cent. Publ. 20, 139-146 (1988).
[For the entire collection see Zbl 0653.00009.]
Let \(H=(f,g):= U\to {\mathbb{C}}^ 2\), \(0\in U\subset {\mathbb{C}}^ 2\), be a holomorphic map with isolated zero at 0. The Łojasiewicz exponent of H at 0 is \[ \lambda (H):=\inf \{\nu \in {\mathbb{R}}| \quad \exists A,B>0,\quad \forall | z| <B,\quad A| z|^{\nu}\leq | H(h)| \}. \] It is well known that the infimum is attained, that \(\lambda\) (H) is a rational number and that there exists an analytic path through 0 on which H has growth order equal to \(\lambda\) (H). The authors prove a precise version of these results; in particular, they determine \(\lambda\) (H) in terms of the multiplicity of f (resp. g) and the intersection multiplicity of f (resp. g) with the different branches of g (resp. f). More precisely: Let \(f=f_ 1\cdot...\cdot f_ r\), \(g=g_ 1\cdot...\cdot g_ s\) the primefactor decomposition of the germs of f and g, \(\mu (f,g_ i)=\dim_{{\mathbb{C}}} O_{{\mathbb{C}}^ 2,0}/(f,g_ i)\) the intersection multiplicity of f and \(g_ i\) and \(ord(g_ i)\) the multiplicity of \(g_ i\). Then
(i) \(\lambda (H)=\max_{i,j}\{\mu (f,g_ i)/ord(g_ i),\mu (g,f_ i)/ord(f_ j)\},\)
(ii) \(H(z)\sim | z|^{\lambda (H)}\) on that branch \(\{g_ i=0\}\) or \(\{f_ j=0\}\) for which the maximum in i) is attained.
For irreducible f and g these results where proved before by Ploski. The proof uses the “horn neighbourhood” method developed by Vuó and Lu in the case of a gradient of a holomorphic function.
Reviewer: G.-M.Greuel

MSC:
32S05 Local complex singularities