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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

##### MSC:
 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
##### Keywords:
analytic curves; intersection multiplicity