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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Łojasiewicz inequality for weighted homogeneous polynomial with isolated singularity
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by Shengli Tan, Stephen S.-T. Yau and Huaiqing Zuo PDF
Proc. Amer. Math. Soc. 138 (2010), 3975-3984 Request permission

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 $|\nabla f(z)|$.
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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
  • ORCID: 0000-0001-6763-1681
  • 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
  • MR Author ID: 185485
  • 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
  • MR Author ID: 872358
  • Email: hqzuo@hotmail.com
  • 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
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 138 (2010), 3975-3984
  • MSC (2010): Primary 32S05; Secondary 14B05
  • DOI: https://doi.org/10.1090/S0002-9939-2010-10387-8
  • MathSciNet review: 2679619