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

 

The Łojasiewicz exponent of a continuous subanalytic function at an isolated zero


Author: Phạm Tiến Sơn
Journal: Proc. Amer. Math. Soc. 139 (2011), 1-9
MSC (2010): Primary 14B05; Secondary 32S05
Published electronically: September 3, 2010
MathSciNet review: 2729065
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Abstract: Let $ f$ be a continuous subanalytic function defined in a neighborhood of the origin $ 0 \in \mathbb{R}^n$ such that $ f$ has an isolated zero at $ 0.$ We describe the smallest possible exponents $ \alpha, \beta, \theta$ for which we have the following estimates: $ \vert f(x)\vert \ge c \Vert x\Vert^\alpha, {\frak m}_f(x) \ge c \Vert x\Vert^\beta, {\frak m}_f(x) \ge c \vert f(x)\vert^\theta$ for $ x$ near zero, where $ c > 0$ and $ {\frak m}_f(x)$ is the nonsmooth slope of $ f$ at $ x.$ We prove that $ \alpha = \beta + 1, \theta = \beta/\alpha.$ In the smooth case, we have $ {\frak m}_f(x) = \Vert\nabla f(x) \Vert,$ and we therefore retrieve a result of Gwoździewicz, which is a counterpart of the result of Teissier in the complex case.


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Additional Information

Phạm Tiến Sơn
Affiliation: Department of Mathematics, University of Dalat, 1, Phu Dong Thien Vuong, Dalat, Vietnam
Email: sonpt@dlu.edu.vn

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10683-4
PII: S 0002-9939(2010)10683-4
Keywords: Łojasiewicz exponent, nonsmooth slope, subanalytic function, subdifferential, tangency variety
Received by editor(s): September 4, 2009
Published electronically: September 3, 2010
Additional Notes: This work was supported by NAFOSTED (Vietnam)
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.