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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Łojasiewicz inequality for weighted homogeneous polynomial with isolated singularity


Authors: Shengli Tan, Stephen S.-T. Yau and Huaiqing Zuo
Journal: Proc. Amer. Math. Soc. 138 (2010), 3975-3984
MSC (2010): Primary 32S05; Secondary 14B05
Published electronically: June 4, 2010
MathSciNet review: 2679619
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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$, $ \vert\nabla f(z)\vert\geq c\vert z\vert^\theta$ in a neighborhood of $ 0,$ which provides the optimal lower estimate of $ \vert\nabla f(z)\vert$.


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Additional Information

Shengli Tan
Affiliation: Department of Mathematics, East China Normal University, No. 500, Dongchuan Road, Shanghai, People’s Republic of China, 200241
Email: sltan@math.ecnu.edu.cn

Stephen S.-T. Yau
Affiliation: Institute of Mathematics, East China Normal University, Shanghai, People’s Republic of China, 200241
Address at time of publication: Department of Mathematics, Statistics, and Computer Science, M/C 249, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, Illinois 60607-7045
Email: yau@uic.edu

Huaiqing Zuo
Affiliation: Department of Mathematics, East China Normal University, No. 500, Dongchuan Road, Shanghai, People’s Republic of China, 200241 – and – Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, Illinois 60607-7045
Email: hqzuo@hotmail.com

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10387-8
PII: S 0002-9939(2010)10387-8
Received by editor(s): October 5, 2009
Received by editor(s) in revised form: January 15, 2010
Published electronically: June 4, 2010
Additional Notes: The first author was supported by NSFC and PSSCS of Shanghai
The second author’s research was partially supported by the NSF
The third author was supported by NSFC and PSSCS of Shanghai
Dedicated: Professor Charles Fefferman on the occasion of his 60th birthday
Communicated by: Mei-Chi Shaw
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.