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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 exponent near the fibre of a mapping


Authors: Tomasz Rodak and Stanisław Spodzieja
Journal: Proc. Amer. Math. Soc. 139 (2011), 1201-1213
MSC (2010): Primary 14R25; Secondary 58K55, 58K05
Published electronically: September 29, 2010
MathSciNet review: 2748414
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Abstract: Let $ g:X\to \mathbb{R}^k$ and $ f:X\to\mathbb{R}^m$, where $ X\subset \mathbb{R}^n$, be continuous semi-algebraic mappings, and $ \lambda\in\mathbb{R}^m$. We describe the optimal exponent $ \theta=:\mathcal{L}_{\infty,f\to\lambda}(g)$ for which the Łojasiewicz inequality $ \vert g(x)\vert\geqslant C\vert x\vert^\theta$ holds with $ C>0$ as $ \vert x\vert\to \infty$ and $ f(x)\to \lambda$. We prove that there exists a semi-algebraic stratification $ \mathbb{R}^m=S_1\cup\cdots\cup S_j$ such that the function $ \lambda\mapsto \mathcal{L}_{\infty,f\to \lambda}(g)$ is constant on each stratum $ S_i$. We apply this result to describe the set of generalized critical values of $ f$.


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Additional Information

Tomasz Rodak
Affiliation: Faculty of Mathematics and Computer Science, University of Łódź, S. Banacha 22, 90-238 Łódź, Poland
Email: rodakt@math.uni.lodz.pl

Stanisław Spodzieja
Affiliation: Faculty of Mathematics and Computer Science, University of Łódź, S. Banacha 22, 90-238 Łódź, Poland
Email: spodziej@math.uni.lodz.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10623-8
PII: S 0002-9939(2010)10623-8
Keywords: Łojasiewicz exponent at infinity, generalized critical values, stratification.
Received by editor(s): May 19, 2009
Received by editor(s) in revised form: April 19, 2010
Published electronically: September 29, 2010
Additional Notes: This research was partially supported by the program POLONIUM
Communicated by: Ted Chinburg
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.