An o-minimal structure which does not admit C ∞ cellular decomposition
[Une structure o-minimale sans décomposition cellulaire lisse]
Le Gal, Olivier ; Rolin, Jean-Philippe
Annales de l'Institut Fourier, Tome 59 (2009), p. 543-562 / Harvested from Numdam
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Nous présentons un exemple de structure o-minimale n’admettant pas la propriété de décomposition cellulaire C ∞ . Pour ce faire, nous construisons une fonction H dont le germe en 0 admet un représentant C k pour tout entier k, mais n’admet aucun représentant C ∞ . Une condition de transcendance sur les coefficients de la série de Taylor de H assure alors la quasi-analyticité de certaines algèbres différentielles 𝒜 n (H) engendrées par H. La o-minimalité de la structure engendrée par H est enfin déduite de cette quasi-analyticité.

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.

Publié le : 2009-01-01
DOI : https://doi.org/10.5802/aif.2439
Classification:  03C64 57-99 26A27 57R45
Mots clés: o-minimal, decomposition cellulaire lisse 
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     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},
     volume = {59},
     year = {2009},
     pages = {543-562},
     doi = {10.5802/aif.2439},
     zbl = {1193.03065},
     mrnumber = {2521427},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2009__59_2_543_0}
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Le Gal, Olivier; Rolin, Jean-Philippe. An o-minimal structure which does not admit $C^{\infty }$ cellular decomposition. Annales de l'Institut Fourier, Tome 59 (2009) pp. 543-562. doi : 10.5802/aif.2439. http://gdmltest.u-ga.fr/item/AIF_2009__59_2_543_0/

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