Théorème de préparation pour les fonctions logarithmico-exponentielles. (Preparation theorem for logarithmico-exponential functions). (French) Zbl 0873.32004

Summary: 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.


32B05 Analytic algebras and generalizations, preparation theorems
32B20 Semi-analytic sets, subanalytic sets, and generalizations
14P15 Real-analytic and semi-analytic sets
Full Text: DOI Numdam EuDML


[1] [A] , Algebraic geometry for scientists and engineers, Amer. Math. Soc., MSM 35 (1990). · Zbl 0709.14001
[2] [DD] , , p-adic and real subanalytic sets, Ann. of Maths, 128 (1988), 79-138. · Zbl 0693.14012
[3] [DMM] , et , The elementary theory of restricted anlytic fields with exponentiation, Annals of Maths, 140 (1994), 183-205. · Zbl 0837.12006
[4] [G] , Complements of subanalytic sets and existential formulas for analytic functions, Inventiones Mathematicae, 125 (1996), 1-12. · Zbl 0851.32009
[5] [H] , An introduction to complex analysis in several variables, North-Holland, 1973. · Zbl 0271.32001
[6] [HLT] , et , Planificateur local en géométrie analytique et aplatissement local, Astérisque, 7-8 (1973), 441-463. · Zbl 0287.14007
[7] [M] , Expansions of the real field with power functions, Ann. Pure Appl. Logic, 68 (1994). · Zbl 0823.03018
[8] [P] , Lipschitz stratification of subanalytic sets, Ann. Scient. École Normale Supérieure, 4e série, 27 (1994), 661-696. · Zbl 0819.32007
[9] [R] , 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. · Zbl 0791.03018
[10] [T] , Paramétrisations de petits chemins en géométrie analytique réelle, preprint, Université de Rennes. · Zbl 0852.32006
[11] [W] , Model completeness results for expansions of real field II: The exponential function, preprint (1991).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.