Théorème de préparation pour les fonctions logarithmico-exponentielles
Lion, Jean-Marie ; Rolin, Jean-Philippe
Annales de l'Institut Fourier, Tome 47 (1997), p. 859-884 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

Nous donnons une preuve géométrique du théorème d’élimination des quantificateurs pour les fonctions logarithmico-exponentielles prouvé initialement par van den Dries, Macintyre et Marker. Notre démonstration n’utilise pas de Théorie des Modèles. Elle repose sur un théorème de préparation pour les fonctions sous-analytiques.

We give a geometric proof of the quantifier elimination theorem for logarithmico-exponential functions, already proved by van den Dries, Macintyre and Marker. Our proof does not make use of model theory arguments. It is based upon a preparation theorem for subanalytic functions.

@article{AIF_1997__47_3_859_0,
     author = {Lion, Jean-Marie and Rolin, Jean-Philippe},
     title = {Th\'eor\`eme de pr\'eparation pour les fonctions logarithmico-exponentielles},
     journal = {Annales de l'Institut Fourier},
     volume = {47},
     year = {1997},
     pages = {859-884},
     doi = {10.5802/aif.1583},
     mrnumber = {98h:32009},
     zbl = {0873.32004},
     language = {fr},
     url = {http://dml.mathdoc.fr/item/AIF_1997__47_3_859_0}
}

Lion, Jean-Marie; Rolin, Jean-Philippe. Théorème de préparation pour les fonctions logarithmico-exponentielles. Annales de l'Institut Fourier, Tome 47 (1997) pp. 859-884. doi : 10.5802/aif.1583. http://gdmltest.u-ga.fr/item/AIF_1997__47_3_859_0/

[A] S.S. Abhyankar, Algebraic geometry for scientists and engineers, Amer. Math. Soc., MSM 35 (1990). | MR 92a:14001 | Zbl 0709.14001

[DD] J. Denef, L. Van Den Dries, p-adic and real subanalytic sets, Ann. of Maths, 128 (1988), 79-138. | MR 89k:03034 | Zbl 0693.14012

[DMM] L. Van Den Dries, A. Macintyre et D. Marker, The elementary theory of restricted anlytic fields with exponentiation, Annals of Maths, 140 (1994), 183-205. | MR 95k:12015 | Zbl 0837.12006

[G] A.M. Gabrielov, Complements of subanalytic sets and existential formulas for analytic functions, Inventiones Mathematicae, 125 (1996), 1-12. | MR 97h:32007 | Zbl 0851.32009

[H] L. Hörmander, An introduction to complex analysis in several variables, North-Holland, 1973. | Zbl 0271.32001

[HLT] H. Hironaka, M. Lejeune-Jalabert et B. Teissier, Planificateur local en géométrie analytique et aplatissement local, Astérisque, 7-8 (1973), 441-463. | Zbl 0287.14007

[M] C. Miller, Expansions of the real field with power functions, Ann. Pure Appl. Logic, 68 (1994). | MR 95i:03081 | Zbl 0823.03018

[P] A. Parusiński, Lipschitz stratification of subanalytic sets, Ann. Scient. École Normale Supérieure, 4e série, 27 (1994), 661-696. | Numdam | MR 96g:32017 | Zbl 0819.32007

[R] J.-P. Ressayre, Integer parts of real closed exponential fields, Arithmetic, Proof Theory and Computational Complexity, P. Clote and J. Krajicek, eds., Oxford University Press (1993), 278-288. | MR 1236467 | Zbl 0791.03018

[T] J.-C. Tougeron, Paramétrisations de petits chemins en géométrie analytique réelle, preprint, Université de Rennes. | Zbl 0852.32006

[W] A.J. Wilkie, Model completeness results for expansions of real field II: The exponential function, preprint (1991).