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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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