On the Łojasiewicz exponent of the gradient of a holomorphic function
Lenarcik, Andrzej
Banach Center Publications, Tome 43 (1998), p. 149-166 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

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 |gradh(x,y)|≥c|(x,y)|λ holds near 0∈C2 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.

Publié le : 1998-01-01
EUDML-ID : urn:eudml:doc:208877 
@article{bwmeta1.element.bwnjournal-article-bcpv44i1p149bwm,
     author = {Lenarcik, Andrzej},
     title = {On the \L ojasiewicz exponent of the gradient of a holomorphic function},
     journal = {Banach Center Publications},
     volume = {43},
     year = {1998},
     pages = {149-166},
     zbl = {0924.32007},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv44i1p149bwm}
}

Lenarcik, Andrzej. On the Łojasiewicz exponent of the gradient of a holomorphic function. Banach Center Publications, Tome 43 (1998) pp. 149-166. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv44i1p149bwm/

[000] [BŁ] J. Bochnak and S. Łojasiewicz, A converse of the Kuiper-Kuo theorem, in: Proceedings of Liverpool Singularities-Symposium I, Lecture Notes in Math. 192, Springer, Berlin, 1971, 254-261. | Zbl 0221.58002

[001] [BK] E. Brieskorn and H. Knörer, Ebene algebraische Kurven, Birkhäuser, Basel, 1981.

[002] [ChK1] J. Chądzyński and T. Krasiński, The Łojasiewicz exponent of an analytic mapping of two complex variables at an isolated zero, in: Singularities, Banach Center Publ. 20, PWN-Polish Science Publishers, Warszawa, 1988, 139-146. | Zbl 0674.32004

[003] [ChK2] J. Chądzyński and T. Krasiński, Resultant and the Łojasiewicz exponent, Ann. Polon. Math. 61 (1995), 95-100. | Zbl 0833.14003

[004] [Fu] T. Fukui, Łojasiewicz type inequalities and Newton diagrams, Proc. Amer. Math. Soc. 112 (1991), 1169-1183. | Zbl 0737.58001

[005] [Kou] A. G. Kouchnirenko, Polyèdres de Newton et nombres de Milnor, Invent. Math. 32 (1976), 1-31.

[006] [Kuch1] W. Kucharz, Examples in the theory of sufficiency of jets, Proc. Amer. Math. Soc. 96 (1986), 163-166. | Zbl 0594.58008

[007] [Kuch2] W. Kucharz, Newton polygons and topological determinancy of analytic germs, Period. Math. Hungar. 22 (1991), 129-132. | Zbl 0743.57019

[008] [Kuip] N. H. Kuiper, C1-equivalence of functions near isolated critical points, in: Symposium on Infinite-Dimensional Topology, R. D. Anderson (ed.), Ann. of Math. Studies 69, Princeton Univ. Press, Princeton, 1972, 199-218.

[009] [K] T. C. Kuo, On C0 sufficiency of jets of potential functions, Topology 8 (1969), 167-171. | Zbl 0183.04601

[010] [KL] T. C. Kuo and Y. C. Lu, On analytic function germ of two complex variables, Topology 16 (1977), 299-310. | Zbl 0378.32001

[011] [LJT] M. Lejeune-Jalabert and B. Teissier, Cloture integral des idéaux et equisingularité, Centre Math. École Polytechnique, Paris, 1974.

[012] [Li] B. Lichtin, Estimation of Łojasiewicz exponent and Newton polygons, Invent. Math. 64 (1981), 417-429. | Zbl 0556.32003

[013] [LCh] Y. C. Lu and S. S. Chang, On C0 sufficiency of complex jets, Canad. J. Math. 25 (1973), 874-880.

[014] [Ł] S. Łojasiewicz, Ensembles semi-analytiques, Inst. de Hautes Études Scientifiques, Bures-sur-Yvette, 1965.

[015] [Pł1] A. Płoski, Une évaluation pour les sous-ensembles analytiques complexes, Bull. Polish Acad. Sci. Math. 31 (1983), 259-262. | Zbl 0578.32013

[016] [Pł2] A. Płoski, Newton polygons and the Łojasiewicz exponent of a holomorphic mapping of C2, Ann. Polon. Math. 51 (1990), 275-281. | Zbl 0764.32012

[017] [Te] B. Teissier, Variétés polaires, Invent. Math. 40 (1977), 267-292. | Zbl 0446.32002