We present an example of an o-minimal structure which does not admit cellular decomposition. To this end, we construct a function whose germ at the origin admits a representative for each integer , but no representative. A number theoretic condition on the coefficients of the Taylor series of then insures the quasianalyticity of some differential algebras induced by . The o-minimality of the structure generated by is deduced from this quasianalyticity property.
Nous présentons un exemple de structure o-minimale n’admettant pas la propriété de décomposition cellulaire . Pour ce faire, nous construisons une fonction dont le germe en admet un représentant pour tout entier , mais n’admet aucun représentant . Une condition de transcendance sur les coefficients de la série de Taylor de assure alors la quasi-analyticité de certaines algèbres différentielles engendrées par . La o-minimalité de la structure engendrée par est enfin déduite de cette quasi-analyticité.
Accepted:
DOI: 10.5802/aif.2439
Classification: 03C64 57-99 26A27 57R45
Keywords: o-minimal, smooth cell decomposition
Author's affiliations:
@article{AIF_2009__59_2_543_0, author = {Le Gal, Olivier and Rolin, Jean-Philippe}, title = {An o-minimal structure which does not admit $C^{\infty }$ cellular decomposition}, journal = {Annales de l'Institut Fourier}, pages = {543--562}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {59}, number = {2}, year = {2009}, doi = {10.5802/aif.2439}, mrnumber = {2521427}, zbl = {1193.03065}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2439/} }
TY - JOUR TI - An o-minimal structure which does not admit $C^{\infty }$ cellular decomposition JO - Annales de l'Institut Fourier PY - 2009 DA - 2009/// SP - 543 EP - 562 VL - 59 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2439/ UR - https://www.ams.org/mathscinet-getitem?mr=2521427 UR - https://zbmath.org/?q=an%3A1193.03065 UR - https://doi.org/10.5802/aif.2439 DO - 10.5802/aif.2439 LA - en ID - AIF_2009__59_2_543_0 ER -
%0 Journal Article %T An o-minimal structure which does not admit $C^{\infty }$ cellular decomposition %J Annales de l'Institut Fourier %D 2009 %P 543-562 %V 59 %N 2 %I Association des Annales de l’institut Fourier %U https://doi.org/10.5802/aif.2439 %R 10.5802/aif.2439 %G en %F AIF_2009__59_2_543_0
Le Gal, Olivier; Rolin, Jean-Philippe. An o-minimal structure which does not admit $C^{\infty }$ cellular decomposition. Annales de l'Institut Fourier, Volume 59 (2009) no. 2, pp. 543-562. doi : 10.5802/aif.2439. https://aif.centre-mersenne.org/articles/10.5802/aif.2439/
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