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An inequality for the intersection multiplicity of analytic curves. (English) Zbl 0699.13014
Let \(f=f_ 1\cdot...\cdot f_ k\) and \(g=g_ 1\cdot...\cdot g_{\ell}\) be decompositions of the power series \(f,g\in {\mathbb{C}}\{x,y\}\) into irreducible factors. Then it is shown that, for each irreducible power series \(h\in {\mathbb{C}}\{x,y\}\), one has \(m_ 0(f,h)/ord(h)\leq \max_{j}\{m_ 0(f,g_ j)/ord(g_ j)\}\) or \(m_ 0(g,h)/ord(h)\leq \max_{i}\{m_ 0(f_ i,g)/ord(f_ i)\}.\)- Here \(m_ 0(f,g)\) denotes the intersection multiplicity of the curves \(\{f=0\}\) and \(\{g=0\}\) and ord(h) stands for the order of the series h. This theorem immediately implies an earlier result of A. Płoski [Bull. Pol. Acad. Sci., Math. 33, 601-605 (1985; Zbl 0606.32001)] and a formula of Płoski and the J. Chądzyński and I. Krasiński [Singularities, Banach Cent. Publ. 20, 139-146 (1988; Zbl 0674.32004)] for the \({\L}ojasiewicz\) exponent \(\ell_ 0(f,g)\).
Reviewer: K.Wolffhardt

13J05 Power series rings
32B05 Analytic algebras and generalizations, preparation theorems
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
13F25 Formal power series rings