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THE ŁOJASIEWICZ EXPONENT FOR WEIGHTED HOMOGENEOUS POLYNOMIAL WITH ISOLATED SINGULARITY

Part of: Singularities Local theory

Published online by Cambridge University Press:  10 June 2016

OULD M. ABDERRAHMANE
OULD M. ABDERRAHMANE*
Affiliation:
Déparement de Mathématiques, Université des Sciences, de Technologie et de Médecine BP. 880, Nouakchott, Mauritanie e-mail: [email protected]
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Abstract

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The purpose of this paper is to give an explicit formula of the Łojasiewicz exponent of an isolated weighted homogeneous singularity in terms of its weights.

MSC classification

Primary: 14B05: Singularities
Secondary: 32S05: Local singularities
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 59 , Issue 2 , May 2017 , pp. 493 - 502
Copyright
Copyright © Glasgow Mathematical Journal Trust 2016 

References

REFERENCES

1. Abderrahmane, O. M., On the Łojasiewicz exponent and Newton polyhedron, Kodai Math. J. 28 (2005), 106–110.Google Scholar
2. Arnold, V. I., Gusein-Zade, S. M. and Varchenko, A. N., Singularities of Differentiable Maps, Vol. 1, Mono- graphs Math., vol. 82 (Birkhäuser, Boston, 1985).Google Scholar
3. Brzostowski, S., The Łojasiewicz Exponent of Semiquasihomogeneous Singularities, http://arxiv.org/abs/1405.5179v1, May 2014.Google Scholar
4. Brzostowski, S., Krasiński, T. and Oleksik, G., A conjecture on the Lojasiewicz exponent, J. Singularities 6 (2012), 124–130.Google Scholar
5. Chadzyński, J. and Krasiński, T., The Łojasiewicz exponent of an analytic mapping of two complex variables at an isolated zero, in Singularities (PWN, Warszawa, 1988), 139–146.Google Scholar
6. Lenarcik, A., On the Łojasiewicz exponent of the gradient of a holomorphic function, in Singularities SymposiumÎ Łojasiewicz 70, Banach Center Publications, vol. 44 (PWN, Warszawa, 1998), 149–166.Google Scholar
7. Fukui, T., Łojasiewicz type inequalities and Newton diagrams, Proc. Amer. Math. Soc. 112 (1991), 1169–1183.Google Scholar
8. Greuel, G.-M., Constant Milnor number implies constant multiplicity for quasihomogeneous singularities, Manuscritpta Math. 56 (1986), 159–166.Google Scholar
9. Krasiński, T., Oleksik, G. and Płoski, A., The Łojasiewicz exponent of an isolated weighted homogeneous surface singularity, Proc. Amer. Math. Soc. 137 (2009), 3387–3397.Google Scholar
10. Kuo, T. C. and Lu, Y. C., On analytic function germs of two complex variables, Topology 16 (1977), 299–310.Google Scholar
11. Lejeune-Jalabert, M. and Teissier, B., Clôture integrale des idéaux et équisingularite, Séminaire Lejeune-Teissier} Centre de Mathématiques, École Polytechnique (Université Scientifique et Medicale de Grenoble, 1974).Google Scholar
12. Lê, D. T. and Ramanujam, C. P., Invariance of Milnor's number implies the invariance of topological type, Amer. J. Math. 98 (1976), 67–78.Google Scholar
13. Lê, D.T. and Saito, K., La constence du nombre de Milnor donne des bonnes stratifications, Compt. Rendus Acad. Sci. Paris, série A 272 (1973), 793–795.Google Scholar
14. Lichtin, B., Estimation of Łojasiewicz exponents and Newton polygons, Invent. Math. 64 (1981), 417–429.Google Scholar
15. Milnor, J., Singular points of complex hypersurfaces (Princeton University Press, Princeton, NJ, 1968).Google Scholar
16. Milnor, J. and Orlik, P., Isolated singularities defined by weighted homogeneous polynomials, Topology 9 (1970), 385–393.Google Scholar
17. O'Shea, D. B., Topologically trivial deformations of isolated quasihomogeneous singularities are equimultiple, Proc. A.M.S. 101 (2) (1987), 260–262.CrossRefGoogle Scholar
18. Paunescu, L., A weighted version of the Kuiper-Kuo-Bochnak-Łojasiewicz theorem, J. Algebr. Geom. 2 (1993), 69–79.Google Scholar
19. Płoski, A., Sur l'exposant d'une application analytique. I, Bull. Polish Acad. Sci. Math. 32 (1984), 669–673.Google Scholar
20. Ploski, A., Semicontinuity of the Lojasiewicz exponent, Univ. Iagel. Acta Math. 48 (2010), 103–110.Google Scholar
21. Tan, S., Yau, S. S.-T. and Zuo, H., Łojasiewicz inequality for weighted homogeneous polynomial with isolated singularity, Proc. Amer. Math. Soc. 138 (2010), 3975–3984.Google Scholar
22. Teissier, B., Variétés polaires, Invent. Math. 40 (1977), 267–292 Google Scholar
23. Varchenko, A. N., A lower bound for the codimension of the stratum μ-constant in term of the mixed Hodge structure, Vest. Mosk. Univ. Mat. 37 (1982), 29–31 Google Scholar