no. 3
Logic
The theory of closed ordered differential fields with m commuting derivations
[La théorie des corps ordonnés différentiellement clos munis de m dérivations commutant entre elles]

Présenté par : Jean-Yves Girard

Rivière, Cédric1
1 Université Denis-Diderot Paris 7, équipe de logique mathématique, 2, place Jussieu, 75251 Paris cedex 05, France
Comptes Rendus. Mathématique, Tome 343 (2006) no. 3, pp. 151-154.

Nous généralisons les travaux de M. Singer concernant la théorie des corps ordonnés différentiellement clos au cas des corps ordonnés munis de m dérivations commutant entre elles. Nous donnons une axiomatisation algébrique de la modèle-complétion de cette théorie et nous pouvons directement déduire que cette dernière admet l'élimination des quantificateurs dans le langage naturel des anneaux ordonnés différentiels.

We generalize the work of M. Singer (1978) on the theory of closed ordered differential fields to the case of m-ODF, the theory of ordered fields equipped with m commuting derivations. We give an algebraic axiomatization of the model completion (denoted by m-CODF) of this theory and we can immediately deduce that m-CODF has quantifier elimination in the natural language of ordered Δ-rings.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2006.06.019
Rivière, Cédric 1

1 Université Denis-Diderot Paris 7, équipe de logique mathématique, 2, place Jussieu, 75251 Paris cedex 05, France 
@article{CRMATH_2006__343_3_151_0,
     author = {Rivi\`ere, C\'edric},
     title = {The theory of closed ordered differential fields with \protect\emph{m} commuting derivations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {151--154},
     publisher = {Elsevier},
     volume = {343},
     number = {3},
     year = {2006},
     doi = {10.1016/j.crma.2006.06.019},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2006.06.019/}
}
TY  - JOUR
AU  - Rivière, Cédric
TI  - The theory of closed ordered differential fields with m commuting derivations
JO  - Comptes Rendus. Mathématique
PY  - 2006
SP  - 151
EP  - 154
VL  - 343
IS  - 3
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2006.06.019/
DO  - 10.1016/j.crma.2006.06.019
LA  - en
ID  - CRMATH_2006__343_3_151_0
ER  - 
%0 Journal Article
%A Rivière, Cédric
%T The theory of closed ordered differential fields with m commuting derivations
%J Comptes Rendus. Mathématique
%D 2006
%P 151-154
%V 343
%N 3
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2006.06.019/
%R 10.1016/j.crma.2006.06.019
%G en
%F CRMATH_2006__343_3_151_0
Rivière, Cédric. The theory of closed ordered differential fields with m commuting derivations. Comptes Rendus. Mathématique, Tome 343 (2006) no. 3, pp. 151-154. doi : 10.1016/j.crma.2006.06.019. http://www.numdam.org/articles/10.1016/j.crma.2006.06.019/

[1] Kolchin, E.R. Differential Algebra and Algebraic Groups, Pure and Applied Mathematics, vol. 54, Academic Press, 1973

[2] Sacks, G. Saturated Model Theory, Benjamin, 1972

[3] Tressl, M. The uniform companion for large differential fields of characteristic zero, Trans. Amer. Math. Soc., Volume 357 (2005), pp. 3933-3951

[4] van den Dries, L.; Schmidt, K. Bounds in the theory of polynomials rings over fields. A nonstandard approach, Invent. Math., Volume 76 (1984), pp. 77-91

Cité par Sources :