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Smooth parameterizations of power-subanalytic sets and compositions of Gevrey functions

Part of: Real functions: Miscellaneous topics Model theory

Published online by Cambridge University Press:  04 June 2021

Siegfried Van Hille
Siegfried Van Hille*
Affiliation:
KU Leuven, Celestijnenlaan 200B, 3001Leuven, Belgium (siegfried.vanhille@kuleuven.be)
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Abstract

We show that if $X$ is an $m$-dimensional definable set in $\mathbb {R}_\text {an}^\text{pow}$, the structure of real subanalytic sets with real power maps added, then for any positive integer $r$ there exists a $C^{r}$-parameterization of $X$ consisting of $cr^{m^{3}}$ maps for some constant $c$. Moreover, these maps are real analytic and this bound is uniform for a definable family.

Keywords

Gevrey functionsmild functionssubanalytic setsCr-parameterizationpreparation of power-subanalytic functionsWeierstrass systems

MSC classification

Primary: 03C98: Applications of model theory 14P15: Real analytic and semianalytic sets 32B20: Semi-analytic sets and subanalytic sets
Secondary: 26E10: $C^infty$-functions, quasi-analytic functions 32B05: Analytic algebras and generalizations, preparation theorems
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 64 , Issue 3 , August 2021 , pp. 532 - 554
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Copyright
Copyright © The Author(s) 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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