Łojasiewicz exponent near the fibre of a mapping
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- by Tomasz Rodak and Stanisław Spodzieja PDF
- Proc. Amer. Math. Soc. 139 (2011), 1201-1213 Request permission
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 $|g(x)|\geqslant C|x|^\theta$ holds with $C>0$ as $|x|\to \infty$ 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 $f$.References
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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
- 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
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 1201-1213
- MSC (2010): Primary 14R25; Secondary 58K55, 58K05
- DOI: https://doi.org/10.1090/S0002-9939-2010-10623-8
- MathSciNet review: 2748414