zbMATH — the first resource for mathematics

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.

32B10 Germs of analytic sets, local parametrization
32S05 Local complex singularities
Full Text: DOI EuDML
[1] Ehlers, F.: Dissertation at University of Bonn (1976) unpublished
[2] Kuo, T.C.: OnC 0 sufficiency of jets of potential functions. Topology.8, 167-171 (1969) · Zbl 0183.04601
[3] Kushnirenko, A.G.: Polyedres de Newton et Nombres de Milnor. Inventiones math.32, 1-32 (1976) · Zbl 0328.32007
[4] Lejeune Jalabert, M., Teissier, B.: Seminar on ?Integral closure of ideals and equisingularity theory?, Fourier Institute, Faculte des Sciences 38402 St. Martin d’Aeres · Zbl 1171.13005
[5] Lu, Y.C., Chang, S.S.: OnC 0 sufficiency of complex jets. Canad. J. Math.25, 874-880 (1973) · Zbl 0258.58004
[6] Mumford, D., Kempf, G. et al. Toroidal Embeddings I, Lecture Notes in Mathematics339, Berlin-Heidelberg-New York: Springer 1973
[7] Teissier, B.: ?Sur diverses conditions numeriques d’equisingularite des familles de courbes et un principe de specialisation de la dependence integrale?, tirage Centre de Mathematiques de l’Ecobe Polytechnique, 91120 Palaiseau, No. 208, 0675 (1975)
[8] Feissier, B.: Varietes Polaires I ? Invariants Polaires des Singulariets d’hypersurfaces. Inventiones math.,40, 267-292 (1977) · Zbl 0446.32002
[9] Varcenko, A.N.: Zeta function of Monodromy and Newton’s Diagram. Inventiones math.37, 253-267 (1977) · Zbl 0434.14017
[10] Vercenko, A.V.: Newton Polyhedra and Estimation of Oscillatory Integrals. Functional Anal. Appl.10, 175-196 (1977) · Zbl 0351.32011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.