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Theorie De Galois Des Equations Differentielles

L. Saidane
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Abstract

Let k be a differential field and C its subfield of constants. In general a differential extension K of k add some new constants to C, and it is difficult to prove that C stay unchangeable under the extension K; This situation is provided by the Picard-Vessiot extension. Kolchin prove the theorem of existence and unicity for these extensions. The aim of this paper is to prove Kolchin theorem and other results, in a simple manner, by means of the theory of models and logic. ----- Soit k un corps différentiel et C son sous corps des constantes. En général une extension différentiel K de k modifie le corps des constantes C de k. Prouver que K ne modifie pas C est un problème assez difficile en algèbre différentiel. Les extensions de Picard-Vessiot constitue un exemple de cette situation. Kolchin a montré le théorème d'existence et d'unicité, à isomorphisme prés, des extensions de Picard-Vessiot sous la condition que le corps C est algébriquement clos. Dans ce travail on utilise la théorie des corps différentiellements clos (Théorie des modèles), pour montrer l'existence et l'unicité, à isomorphisme prés, des extensions de Picard-Vessiot. On calcul ensuite le groupe de Galois différentiel de certaines extensions particulière. Enfin, on montre quelque théorèmes généraux de la théorie de Galois différentielle par les mêmes techniques.

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L. Saidane. 2003-06-10. Theorie De Galois Des Equations Differentielles. https://arxiv.org/abs/math/0306169

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