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Estimation of Lojasiewicz exponents and Newton polygons. (English) Zbl 0556.32003
The author calculates the \(C^ 0\) degree of sufficiency \(\nu_ f\) for an analytic map germ \(f:({\mathbb{C}}^ 2,0)\to ({\mathbb{C}},0)\) satisfying a certain nondegeneracy condition. Thus, \(\nu_ f\) is the least integer \(\nu\) with the property that \(j^{\nu}+f\) is topologically equivalent to \(j^{\nu}f\) for any g of order \(\geq \nu +1\), where \(j^{\nu}f\) is the \(\nu\)-jet of f at 0. The nondegeneracy condition is stronger than A. N. Varchenko’s [Invent. Math. 37, 253-262 (1976; Zbl 0333.14007)], but, as the author proves, it is still generic (satisfied by an open dense set of analytic germs having a given Newton polygon). The proof uses toroidal modifications, and the nondegeneracy condition is needed at one point in order for the calculation to proceed. This is the result: \(\nu_ f\) is at most one more than the largest finite axis intercept of any of the finitely many support lines which meet the Newton polygon (which possibly contains noncompact edges) in an edge.

MSC:
32B10 Germs of analytic sets, local parametrization
32S05 Local complex singularities
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References:
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