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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
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