The theory of closed ordered differential fields with <i>m</i> commuting derivations

Cédric Rivière

doi:10.1016/j.crma.2006.06.019

Logic

The theory of closed ordered differential fields with m commuting derivations

Presented by: Jean-Yves Girard

Cédric Rivière ¹

¹ Université Denis-Diderot Paris 7, équipe de logique mathématique, 2, place Jussieu, 75251 Paris cedex 05, France

Comptes Rendus. Mathématique, Volume 343 (2006) no. 3, pp. 151-154.

Abstract (Original language)
Résumé

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.

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.

Received: 2006-06-06
Accepted: 2006-06-13
Published online: 2006-07-07

DOI: 10.1016/j.crma.2006.06.019

Author's affiliations:

Cédric Rivière 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 = {C\'edric Rivi\`ere},
     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},
}

TY  - JOUR
AU  - Cédric Rivière
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
DO  - 10.1016/j.crma.2006.06.019
LA  - en
ID  - CRMATH_2006__343_3_151_0
ER  -

%0 Journal Article
%A Cédric Rivière
%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
%R 10.1016/j.crma.2006.06.019
%G en
%F CRMATH_2006__343_3_151_0

Cédric Rivière. The theory of closed ordered differential fields with m commuting derivations. Comptes Rendus. Mathématique, Volume 343 (2006) no. 3, pp. 151-154. doi : 10.1016/j.crma.2006.06.019. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.06.019/

References
Cited by

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

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

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

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

Cited by Sources:

Comments - Policy