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We prove that the expansion of the real field by a restricted C${}^{\infty }$-function is generically o-minimal. Such a result was announced by A. Grigoriev, and proved in a different way. Here, we deduce quasi-analyticity from a transcendence condition on Taylor expansions. This then implies o-minimality. The transcendance condition is shown to be generic. As a corollary, we recover in a simple way that there exist o-minimal structures that doesn’t admit analytic cell decomposition, and that there exist incompatible...

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.