Łojasiewicz inequality for weighted homogeneous polynomial with isolated singularity

Tan, Shengli; Yau, Stephen; Zuo, Huaiqing

doi:10.1090/S0002-9939-2010-10387-8

Łojasiewicz inequality for weighted homogeneous polynomial with isolated singularity
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by Shengli Tan, Stephen S.-T. Yau and Huaiqing Zuo

Proc. Amer. Math. Soc. 138 (2010), 3975-3984

DOI: https://doi.org/10.1090/S0002-9939-2010-10387-8

Published electronically: June 4, 2010

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Abstract:

Let $\nabla f$ be a gradient vector field of a weighted homogenous polynomial with isolated critical point at the origin. Let $(w_1,\dotsc ,w_n)$ be the weights of $f$. In this paper, we prove that the Łojasiewicz Exponent $\theta$ of $f$ is precisely equal to $\displaystyle {\max _{0\leq i\leq n}}w_i-1$. This means that for some constant $c$, $|\nabla f(z)|\geq c|z|^\theta$ in a neighborhood of $0,$ which provides the optimal lower estimate of $|\nabla f(z)|$.

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
W. Dale Brownawell, Local Diophantine Nullstellen inequalities, J. Amer. Math. Soc. 1 (1988), no. 2, 311–322. MR 928261, DOI 10.1090/S0894-0347-1988-0928261-3
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
Alexandru Dimca, Topics on real and complex singularities, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1987. An introduction. MR 1013785, DOI 10.1007/978-3-663-13903-4
Igor Dolgachev, Weighted projective varieties, Group actions and vector fields (Vancouver, B.C., 1981) Lecture Notes in Math., vol. 956, Springer, Berlin, 1982, pp. 34–71. MR 704986, DOI 10.1007/BFb0101508
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
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
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
Zbigniew Jelonek, On the effective Nullstellensatz, Invent. Math. 162 (2005), no. 1, 1–17. MR 2198324, DOI 10.1007/s00222-004-0434-8
Shanyu Ji, János Kollár, and Bernard Shiffman, A global Łojasiewicz inequality for algebraic varieties, Trans. Amer. Math. Soc. 329 (1992), no. 2, 813–818. MR 1046016, DOI 10.1090/S0002-9947-1992-1046016-6
Tadeusz Krasiński, Grzegorz Oleksik, and Arkadiusz Płoski, The Łojasiewicz exponent of an isolated weighted homogeneous surface singularity, Proc. Amer. Math. Soc. 137 (2009), no. 10, 3387–3397. MR 2515408, DOI 10.1090/S0002-9939-09-09935-3
Lejeune-Jalabert, M. and Teissier, B., Clôture integrale des idéaux et équisingularite, in: Séminaire Lejeune-Teissier, Centre de Mathématiques, École Polytechnique, Université Scientifique et Medicale de Grenoble, 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
S. Łojasiewicz, Sur le problème de la division, Studia Math. 18 (1959), 87–136 (French). MR 107168, DOI 10.4064/sm-18-1-87-136
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. II, Bull. Polish Acad. Sci. Math. 33 (1985), no. 3-4, 123–127 (French, with English and Russian summaries). MR 805025
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