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

Abstract

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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.

How to cite

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Le 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

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  1. Coste (M.).— An introduction to o-minimal geometry. Instituti editoriali e poligrafici in-ternazionali (2000). 
  2. Denef (J.) and van den Dries (L.).— P-adic and real subanalytic sets. Ann. Math., 128:79-138 (1988). Zbl0693.14012MR951508
  3. Gabrielov (A.).— Complements of subanalytic sets and existential formulas for analytic functions. Invent. math., 125:1-12 (1996). Zbl0851.32009MR1389958
  4. Grigoriev (A.).— On o-minimality of extensions of the real field by restricted generic smooth functions. arXiv.org:math/0506109 (2005). 
  5. Le Gal O. and Rolin (J.-P.).— An o-minimal structure which does not admit C ∞ -cellular decomposition. Ann. Inst. Fourier, Grenoble, 59, 2:543-546 (2008). Zbl1193.03065
  6. 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
  7. 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
  8. Wilkie (A.).— A theorem of the complement and some new o-minimal structures. Sel. math., 5:397-421 (1999). Zbl0948.03037MR1740677

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