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Newton polygons and the Łojasiewicz exponent of a holomorphic mapping of $$\mathbb{C}^ 2$$. (English) Zbl 0764.32012
The Łojasiewicz exponent $\inf\{\theta>0:\max\{| f(z)|,\;| g(z)|\}\geq C| z|^ \theta\text{ for } z\in\mathbb{C}^ 2 \text{ near the origin}$ is estimated in terms of the Newton polygons of $$f$$ and $$g$$ for holomorphic mappings $$(f,g):\mathbb{C}^ 2\to\mathbb{C}^ 2$$ which are convenients, i.e. $$f(0,0)=g(0,0)=0$$ and $$f(X,0)f(0,Y)\neq 0$$, $$g(X,0)g(0,Y)\neq 0$$. For any power series $$f(X,Y)$$, $$g(X,Y)$$ without constant term, the number $$l_ 0(f,g,X)$$ is defined to be $\inf\{\delta>0:\max\{| f(x,y)|,| g(x,y)|\}\geq C| x|^ \delta\text{ for } (x,y)\text{ near } 0\in\mathbb{C}^ 2.$ Analogously, the number $$l_ 0(f,g,Y)$$ is defined. Denoting by $$l_ 0(f,g)$$ the Łojasiewicz exponent we have $l_ 0(f,g)=\max\{l_ 0(f,g,X),l_ 0(f,g,Y)\}.$ A formula for $$l_ 0(f,g,X)$$, and $$l_ 0(f,g,Y)$$ respectively, is obtained too. It is in fact a modification of a result of Chadzyński and Krasiński, proved here independently.
Reviewer: S.Dimiev (Sofia)

##### MSC:
 32S05 Local complex singularities 14B05 Singularities in algebraic geometry
##### Keywords:
holomorphic mapping; Newton polygon; Łojasiewicz exponent
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