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

Le Gal, Olivier; Rolin, Jean-Philippe

Annales de l'Institut Fourier, Tome 59 (2009) no. 2, pp. 543-562.

Nous présentons un exemple de structure o-minimale n’admettant pas la propriété de décomposition cellulaire $C^{\infty }$. 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 $C^{\infty }$. Une condition de transcendance sur les coefficients de la série de Taylor de $H$ assure alors la quasi-analyticité de certaines algèbres différentielles ${𝒜}_n\left(H\right)$ 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 $C^{\infty }$ 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 ${𝒜}_n\left(H\right)$ induced by $H$. The o-minimality of the structure generated by $H$ is deduced from this quasianalyticity property.

Classification : 03C64 57-99 26A27 57R45
Keywords: o-minimal, smooth cell decomposition
Mot clés : o-minimal, decomposition cellulaire lisse

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