On the Łojasiewicz exponent of the gradient of a holomorphic function

Andrzej Lenarcik

Banach Center Publications (1998)

  • Volume: 44, Issue: 1, page 149-166
  • ISSN: 0137-6934

Abstract

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The Łojasiewicz exponent of the gradient of a convergent power series h(X,Y) with complex coefficients is the greatest lower bound of the set of λ > 0 such that the inequality | g r a d h ( x , y ) | ≥ c | ( x , y ) | λ holds near 0 ∈ C 2 for a certain c > 0. In the paper, we give an estimate of the Łojasiewicz exponent of grad h using information from the Newton diagram of h. We obtain the exact value of the exponent for non-degenerate series.

How to cite

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Lenarcik, Andrzej. "On the Łojasiewicz exponent of the gradient of a holomorphic function." Banach Center Publications 44.1 (1998): 149-166. <http://eudml.org/doc/208877>.

@article{Lenarcik1998,
abstract = {The Łojasiewicz exponent of the gradient of a convergent power series h(X,Y) with complex coefficients is the greatest lower bound of the set of λ > 0 such that the inequality $|grad h(x,y)| ≥ c|(x,y)|^λ$ holds near $0 ∈ C^2$ for a certain c > 0. In the paper, we give an estimate of the Łojasiewicz exponent of grad h using information from the Newton diagram of h. We obtain the exact value of the exponent for non-degenerate series.},
author = {Lenarcik, Andrzej},
journal = {Banach Center Publications},
keywords = {Ł ojasiewicz exponent; Newton diagram},
language = {eng},
number = {1},
pages = {149-166},
title = {On the Łojasiewicz exponent of the gradient of a holomorphic function},
url = {http://eudml.org/doc/208877},
volume = {44},
year = {1998},
}

TY - JOUR
AU - Lenarcik, Andrzej
TI - On the Łojasiewicz exponent of the gradient of a holomorphic function
JO - Banach Center Publications
PY - 1998
VL - 44
IS - 1
SP - 149
EP - 166
AB - The Łojasiewicz exponent of the gradient of a convergent power series h(X,Y) with complex coefficients is the greatest lower bound of the set of λ > 0 such that the inequality $|grad h(x,y)| ≥ c|(x,y)|^λ$ holds near $0 ∈ C^2$ for a certain c > 0. In the paper, we give an estimate of the Łojasiewicz exponent of grad h using information from the Newton diagram of h. We obtain the exact value of the exponent for non-degenerate series.
LA - eng
KW - Ł ojasiewicz exponent; Newton diagram
UR - http://eudml.org/doc/208877
ER -

References

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  12. [LJT] M. Lejeune-Jalabert and B. Teissier, Cloture integral des idéaux et equisingularité, Centre Math. École Polytechnique, Paris, 1974. 
  13. [Li] B. Lichtin, Estimation of Łojasiewicz exponent and Newton polygons, Invent. Math. 64 (1981), 417-429. Zbl0556.32003
  14. [LCh] Y. C. Lu and S. S. Chang, On C 0 sufficiency of complex jets, Canad. J. Math. 25 (1973), 874-880. 
  15. [Ł] S. Łojasiewicz, Ensembles semi-analytiques, Inst. de Hautes Études Scientifiques, Bures-sur-Yvette, 1965. 
  16. [Pł1] A. Płoski, Une évaluation pour les sous-ensembles analytiques complexes, Bull. Polish Acad. Sci. Math. 31 (1983), 259-262. Zbl0578.32013
  17. [Pł2] A. Płoski, Newton polygons and the Łojasiewicz exponent of a holomorphic mapping of C 2 , Ann. Polon. Math. 51 (1990), 275-281. Zbl0764.32012
  18. [Te] B. Teissier, Variétés polaires, Invent. Math. 40 (1977), 267-292. Zbl0446.32002

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