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 o-minimal structures. We even obtain o-minimal structures that are not compatible with restricted analytic functions.
On montre que génériquement, l’expansion du corps des réels par une fonction C 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.
@article{AFST_2010_6_19_3-4_479_0,
author = {Olivier Le Gal},
title = {A generic condition implying o-minimality for restricted {C}$^{\infty }$-functions},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {479--492},
publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
address = {Toulouse},
volume = {Ser. 6, 19},
number = {3-4},
year = {2010},
doi = {10.5802/afst.1252},
mrnumber = {2790804},
zbl = {1215.26012},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1252/}
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Olivier Le Gal. A generic condition implying o-minimality for restricted C$^{\infty }$-functions. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 19 (2010) no. 3-4, pp. 479-492. doi : 10.5802/afst.1252. https://afst.centre-mersenne.org/articles/10.5802/afst.1252/
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