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A generic condition implying o-minimality for restricted C -functions

Olivier Le Gal — 2010

Annales de la faculté des sciences de Toulouse Mathématiques

We prove that the expansion of the real field by a restricted C -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...

An o-minimal structure which does not admit C cellular decomposition

Olivier Le GalJean-Philippe Rolin — 2009

Annales de l’institut Fourier

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.

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