A generic condition implying o-minimality for restricted C${}^{\infty }$-functions

Le Gal, Olivier

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 19 (2010) no. 3-4, pp. 479-492.

On montre que génériquement, l’expansion du corps des réels par une fonction C${}^{\infty }$ restreinte est o-minimale. Un résultat du même type utilisant d’autres d’arguments a été annoncé par A. Grigoriev. Ici, nous utilisons une condition de transcendance sur les développements de Taylor pour assurer la quasianalyticité de certaines algèbres différentielles, ce qui implique la o-minimalité. On montre que cette condition de transcendance est générique. Comme corollaire de ce résultat, on donne des preuves simples du fait qu’il existe des structures o-minimales n’admettant pas de décomposition cellulaire analytique, et qu’il existe des structures o-minimales incompatibles. On obtient même des structures o-minimales non compatibles avec les fonctions analytiques restreintes.

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 o-minimal structures. We even obtain o-minimal structures that are not compatible with restricted analytic functions.

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