The Łojasiewicz exponent of an isolated weighted homogeneous surface singularity
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- by Tadeusz Krasiński, Grzegorz Oleksik and Arkadiusz Płoski PDF
- Proc. Amer. Math. Soc. 137 (2009), 3387-3397 Request permission
Abstract:
We give an explicit formula for the Łojasiewicz exponent of an isolated weighted homogeneous surface singularity in terms of its weights. From the formula we get that the Łojasiewicz exponent is a topological invariant of these singularities.References
- Ould M. Abderrahmane, On the Lojasiewicz exponent and Newton polyhedron, Kodai Math. J. 28 (2005), no. 1, 106–110. MR 2122194, DOI 10.2996/kmj/1111588040
- Jacek Chądzyński and Tadeusz Krasiński, The Łojasiewicz exponent of an analytic mapping of two complex variables at an isolated zero, Singularities (Warsaw, 1985) Banach Center Publ., vol. 20, PWN, Warsaw, 1988, pp. 139–146. MR 1101835
- J. Chądzyński and T. Krasiński, Resultant and the Łojasiewicz exponent, Ann. Polon. Math. 61 (1995), no. 1, 95–100. MR 1318321, DOI 10.4064/ap-61-1-95-100
- Toshizumi Fukui, Łojasiewicz type inequalities and Newton diagrams, Proc. Amer. Math. Soc. 112 (1991), no. 4, 1169–1183. MR 1065945, DOI 10.1090/S0002-9939-1991-1065945-5
- Haraux, A. and Pham, T. S.: On the Łojasiewicz exponents of quasi-homogeneous functions. Preprints of the Laboratoire Jacques-Louis Lions 2007, Université Pierre et Marie Curie, No. R07041 (http://www.ann.jussieu.fr/publications/2007/R07041.pdf).
- Tzee Char Kuo and Yung Chen Lu, On analytic function germs of two complex variables, Topology 16 (1977), no. 4, 299–310. MR 460711, DOI 10.1016/0040-9383(77)90037-4
- Lejeune-Jalabert, M. and Teissier, B.: Cloture integrale des idéaux et equisingularité. École Polytechnique, 1974.
- Andrzej Lenarcik, On the Łojasiewicz exponent of the gradient of a holomorphic function, Singularities Symposium—Łojasiewicz 70 (Kraków, 1996; Warsaw, 1996) Banach Center Publ., vol. 44, Polish Acad. Sci. Inst. Math., Warsaw, 1998, pp. 149–166. MR 1677363
- Ben Lichtin, Estimation of Lojasiewicz exponents and Newton polygons, Invent. Math. 64 (1981), no. 3, 417–429. MR 632982, DOI 10.1007/BF01389274
- John Milnor and Peter Orlik, Isolated singularities defined by weighted homogeneous polynomials, Topology 9 (1970), 385–393. MR 293680, DOI 10.1016/0040-9383(70)90061-3
- Peter Orlik and Philip Wagreich, Isolated singularities of algebraic surfaces with C$^{\ast }$ action, Ann. of Math. (2) 93 (1971), 205–228. MR 284435, DOI 10.2307/1970772
- Arkadiusz Płoski, Sur l’exposant d’une application analytique. I, Bull. Polish Acad. Sci. Math. 32 (1984), no. 11-12, 669–673 (French, with English and Russian summaries). MR 786190
- Arkadiusz Płoski, Sur l’exposant d’une application analytique. I, Bull. Polish Acad. Sci. Math. 32 (1984), no. 11-12, 669–673 (French, with English and Russian summaries). MR 786190
- Kyoji Saito, Quasihomogene isolierte Singularitäten von Hyperflächen, Invent. Math. 14 (1971), 123–142 (German). MR 294699, DOI 10.1007/BF01405360
- B. Teissier, Variétés polaires. I. Invariants polaires des singularités d’hypersurfaces, Invent. Math. 40 (1977), no. 3, 267–292 (French). MR 470246, DOI 10.1007/BF01425742
- Stephen S.-T. Yau, Topological types and multiplicities of isolated quasihomogeneous surface singularities, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 447–454. MR 935021, DOI 10.1090/S0273-0979-1988-15695-9
Additional Information
- Tadeusz Krasiński
- Affiliation: Faculty of Mathematics and Computer Science, University of Łódź, Banacha 22, 90-238 Łódź, Poland
- Email: krasinsk@uni.lodz.pl
- Grzegorz Oleksik
- Affiliation: Faculty of Mathematics and Computer Science, University of Łódź, Banacha 22, 90-238 Łódź, Poland
- Email: oleksig@math.uni.lodz.pl
- Arkadiusz Płoski
- Affiliation: Department of Mathematics, Technical University, Al. 1000 LPP 7, 25-314 Kielce, Poland
- Email: matap@tu.kielce.pl
- Received by editor(s): July 10, 2008
- Received by editor(s) in revised form: February 10, 2009
- Published electronically: May 6, 2009
- Communicated by: Mei-Chi Shaw
- © Copyright 2009 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 137 (2009), 3387-3397
- MSC (2000): Primary 32S05
- DOI: https://doi.org/10.1090/S0002-9939-09-09935-3
- MathSciNet review: 2515408