Estimation of Lojasiewicz exponents and Newton polygons
type: Item
instance of: scholarly article
title: Estimation of Lojasiewicz exponents and Newton polygons
zbMATH Open document ID: 0556.32003
DOI: 10.1007/BF01389274
author: Ben Lichtin
published in: Inventiones Mathematicae
publication date: 1981-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/142829
review text: 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 \textit{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.
Mathematics Subject Classification ID: 32B10
Mathematics Subject Classification ID: 32S05
zbMATH DE Number: 3886254
zbMATH Keywords: degree of sufficiency
zbMATH Keywords: analytic map germ
zbMATH Keywords: Newton polygon
zbMATH Keywords: toroidal modifications
MaRDI profile type: Publication
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cites work: Variétés polaires. I: Invariants polaires des singularites d'hypersurfaces
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description: scientific article; zbMATH DE number 3886254