# An o-minimal structure which does not admit ${C}^{\infty}$ 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

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topLe 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 -

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