Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The Łojasiewicz exponent of an isolated weighted homogeneous surface singularity


Authors: Tadeusz Krasinski, Grzegorz Oleksik and Arkadiusz Płoski
Journal: Proc. Amer. Math. Soc. 137 (2009), 3387-3397
MSC (2000): Primary 32S05
DOI: https://doi.org/10.1090/S0002-9939-09-09935-3
Published electronically: May 6, 2009
MathSciNet review: 2515408
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [A] Abderrahmane, O. M.: On the Łojasiewicz exponent and Newton polyhedron. Kodai. Math. J. 28 (2005), 106-110. MR 2122194 (2006c:32031)
  • [CK1] Chadzyński, J. and Krasiński, T.: The Łojasiewicz exponent of an analytic mapping of two complex variables at an isolated zero. In: Singularities, Banach Center Publ. 20, PWN, Warszawa, 1988, 139-146. MR 1101835 (92e:32018)
  • [CK2] Chadzyński, J. and Krasiński, T.: Resultant and the Łojasiewicz exponent. Ann. Polon. Math. 61 (1995), 95-100. MR 1318321 (96a:32061)
  • [F] Fukui, T.: Łojasiewicz type inequalities and Newton diagrams. Proc. Amer. Math. Soc. 112 (1991), 1169-1183. MR 1065945 (91j:58021)
  • [HP] Haraux, A. and Pham, T. S.: On the Ł ojasiewicz 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).
  • [KL] Kuo, T. C. and Lu, Y. C.: On analytic function germs of two complex variables. Topology 16 (1977), 299-310. MR 0460711 (57:704)
  • [L-JT] Lejeune-Jalabert, M. and Teissier, B.: Cloture integrale des idéaux et equisingularité. École Polytechnique, 1974.
  • [L] Lenarcik, A.: On the Łojasiewicz exponent of the gradient of a holomorphic function. In: Singularities Symposium-Łojasiewicz 70. Banach Center Publ. 44, PWN, Warszawa, 1998, 149-166. MR 1677363 (2000h:32039)
  • [Lt] Lichtin, B.: Estimation of Łojasiewicz exponents and Newton polygons. Inventiones Math. 64 (1981), 417-429. MR 632982 (83b:32006)
  • [MO] Milnor, J. and Orlik, P.: Isolated singularities defined by weighted homogeneous polynomials, Topology 9 (1970), 385-393. MR 0293680 (45:2757)
  • [OW] Orlik, P. and Wagreich, P.: Isolated singularities of algebraic surfaces with $ \mathbb{C}^{\ast}$-action. Ann. of Math. 93 (1971), 205-228. MR 0284435 (44:1662)
  • [P1] Płoski, A.: Sur l'exposant d'une application analytique I. Bull. Polish Acad. Sci. Math. 32 (1984), 669-673. MR 786190 (86j:32025a)
  • [P2] Płoski, A.: Sur l'exposant d'une application analytique II. Bull. Polish Acad. Sci. Math. 33 (1985), 123-127. MR 805025 (86j:32025b)
  • [S] Saito, K.: Quasihomogene isolierte Singularitäten von Hyperflächen. Invent. Math. 14 (1971), 123-142. MR 0294699 (45:3767)
  • [T] Teissier, B.: Variétés polaires. Invent. Math. 40 (1977), 267-292. MR 0470246 (57:10004)
  • [Y] Yau, S. S.-T.: Topological types and multiplicities of isolated quasihomogeneous surface singularities. Bull. Amer. Math. Soc. (N.S.) 19 (1988), 447-454. MR 935021 (92b:32042)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 32S05

Retrieve articles in all journals with MSC (2000): 32S05


Additional Information

Tadeusz Krasinski
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

DOI: https://doi.org/10.1090/S0002-9939-09-09935-3
Keywords: Łojasiewicz exponent, weighted homogeneous polynomial, isolated surface singularity, polar curve.
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
Article copyright: © Copyright 2009 American Mathematical Society

American Mathematical Society