An o-minimal structure which does not admit C ∞ cellular decomposition

Olivier Le Gal[1]; Jean-Philippe Rolin[2]

  • [1] University of Toronto Department of Mathematics Toronto, Ontario M5S 2E4 (Canada)
  • [2] Université de Bourgogne IMB, UFR Sciences et Techniques 9, Avenue Alain Savary BP 47870 21078 Dijon (France)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 2, page 543-562
  • ISSN: 0373-0956

Abstract

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We present an example of an o-minimal structure which does not admit C ∞ cellular decomposition. To this end, we construct a function H whose germ at the origin admits a C k representative for each integer k , but no C ∞ representative. A number theoretic condition on the coefficients of the Taylor series of H then insures the quasianalyticity of some differential algebras 𝒜 n ( H ) induced by H . The o-minimality of the structure generated by H is deduced from this quasianalyticity property.

How to cite

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Le Gal, Olivier, and Rolin, Jean-Philippe. "An o-minimal structure which does not admit $C^{\infty }$ cellular decomposition." Annales de l’institut Fourier 59.2 (2009): 543-562. <http://eudml.org/doc/10403>.

@article{LeGal2009,
abstract = {We present an example of an o-minimal structure which does not admit $C^\{\infty \}$ cellular decomposition. To this end, we construct a function $H$ whose germ at the origin admits a $C^k$ representative for each integer $k$, but no $C^\{\infty \}$ representative. A number theoretic condition on the coefficients of the Taylor series of $H$ then insures the quasianalyticity of some differential algebras $\mathcal\{A\}_n(H)$ induced by $H$. The o-minimality of the structure generated by $H$ is deduced from this quasianalyticity property.},
affiliation = {University of Toronto Department of Mathematics Toronto, Ontario M5S 2E4 (Canada); Université de Bourgogne IMB, UFR Sciences et Techniques 9, Avenue Alain Savary BP 47870 21078 Dijon (France)},
author = {Le Gal, Olivier, Rolin, Jean-Philippe},
journal = {Annales de l’institut Fourier},
keywords = {o-minimal; smooth cell decomposition},
language = {eng},
number = {2},
pages = {543-562},
publisher = {Association des Annales de l’institut Fourier},
title = {An o-minimal structure which does not admit $C^\{\infty \}$ cellular decomposition},
url = {http://eudml.org/doc/10403},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Le Gal, Olivier
AU - Rolin, Jean-Philippe
TI - An o-minimal structure which does not admit $C^{\infty }$ cellular decomposition
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 2
SP - 543
EP - 562
AB - We present an example of an o-minimal structure which does not admit $C^{\infty }$ cellular decomposition. To this end, we construct a function $H$ whose germ at the origin admits a $C^k$ representative for each integer $k$, but no $C^{\infty }$ representative. A number theoretic condition on the coefficients of the Taylor series of $H$ then insures the quasianalyticity of some differential algebras $\mathcal{A}_n(H)$ induced by $H$. The o-minimality of the structure generated by $H$ is deduced from this quasianalyticity property.
LA - eng
KW - o-minimal; smooth cell decomposition
UR - http://eudml.org/doc/10403
ER -

References

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