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

Olivier Le Gal^{[1]}

- [1] Dpto. Algebra, Geometría y Topologia. Facultad de Ciencas. E-47005-Valladolid. Spain

Annales de la faculté des sciences de Toulouse Mathématiques (2010)

- Volume: 19, Issue: 3-4, page 479-492
- ISSN: 0240-2963

## Access Full Article

top## Abstract

top## How to cite

topLe Gal, Olivier. "A generic condition implying o-minimality for restricted C$^{\infty }$-functions." Annales de la faculté des sciences de Toulouse Mathématiques 19.3-4 (2010): 479-492. <http://eudml.org/doc/115893>.

@article{LeGal2010,

abstract = {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.},

affiliation = {Dpto. Algebra, Geometría y Topologia. Facultad de Ciencas. E-47005-Valladolid. Spain},

author = {Le Gal, Olivier},

journal = {Annales de la faculté des sciences de Toulouse Mathématiques},

keywords = { functions; restricted strongly transcendental functions; generically o-minimal structure; polynomially bounded set},

language = {eng},

number = {3-4},

pages = {479-492},

publisher = {Université Paul Sabatier, Toulouse},

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

url = {http://eudml.org/doc/115893},

volume = {19},

year = {2010},

}

TY - JOUR

AU - Le Gal, Olivier

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

JO - Annales de la faculté des sciences de Toulouse Mathématiques

PY - 2010

PB - Université Paul Sabatier, Toulouse

VL - 19

IS - 3-4

SP - 479

EP - 492

AB - 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.

LA - eng

KW - functions; restricted strongly transcendental functions; generically o-minimal structure; polynomially bounded set

UR - http://eudml.org/doc/115893

ER -

## References

top- Coste (M.).— An introduction to o-minimal geometry. Instituti editoriali e poligrafici in-ternazionali (2000).
- Denef (J.) and van den Dries (L.).— P-adic and real subanalytic sets. Ann. Math., 128:79-138 (1988). Zbl0693.14012MR951508
- Gabrielov (A.).— Complements of subanalytic sets and existential formulas for analytic functions. Invent. math., 125:1-12 (1996). Zbl0851.32009MR1389958
- Grigoriev (A.).— On o-minimality of extensions of the real field by restricted generic smooth functions. arXiv.org:math/0506109 (2005).
- Le Gal O. and Rolin (J.-P.).— An o-minimal structure which does not admit C${}_{\infty}$-cellular decomposition. Ann. Inst. Fourier, Grenoble, 59, 2:543-546 (2008). Zbl1193.03065
- Rolin J.-P., Speissegger (P.), and Wilkie (A. J.).— Quasianalytic Denjoy-Carleman classes and o-minimality. J. Amer. Math. Soc., 16(4):751-777 (electronic) (2003). Zbl1095.26018MR1992825
- van den Dries (L.).— Tame topology and o-minimal structures, volume 248 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1998). Zbl0953.03045MR1633348
- Wilkie (A.).— A theorem of the complement and some new o-minimal structures. Sel. math., 5:397-421 (1999). Zbl0948.03037MR1740677

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.