[Invariant valuations for differential Galois group action.]
Let be a differential field extension with differential Galois group . For the natural action of G on the Riemann–Zariski variety of the field extension , we study the invariant valuations when they do exist. We show close relations between these invariant valuations and the elements of F holonomic over K. Next, we study the continuity of the derivation ∂ with respect to these ν-adic topologies. We give a geometric structure property of G-invariant valuation inspired by Zariski. Finally, we give an answer for the existence problem of invariant valuations in the context of Picard–Vessiot extension.
Soit une extension de corps différentiels, de groupe de Galois différentiel . Pour l'action naturelle de G sur la variété de Riemann–Zariski de l'extension , nous étudions les valuations invariantes quand elles existent. Nous exhibons les relations entre ces valuations invariantes et les éléments de F holonomes sur K. Puis, nous examinons la continuité de la dérivation ∂ par rapport aux topologies ν-adiques. Nous donnons une propriété de structure géométrique des valuations invariantes, inspirée d'un résultat de Zariski. Enfin nous répondons au problème de l'existence des valuations invariantes dans le contexte des extensions de Picard–Vessiot.
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Guillaume Duval 1
@article{CRMATH_2004__339_11_763_0, author = {Guillaume Duval}, title = {Valuations invariantes pour l'action des groupes de {Galois} diff\'erentiels}, journal = {Comptes Rendus. Math\'ematique}, pages = {763--768}, publisher = {Elsevier}, volume = {339}, number = {11}, year = {2004}, doi = {10.1016/j.crma.2004.05.021}, language = {fr}, }
Guillaume Duval. Valuations invariantes pour l'action des groupes de Galois différentiels. Comptes Rendus. Mathématique, Volume 339 (2004) no. 11, pp. 763-768. doi : 10.1016/j.crma.2004.05.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.05.021/
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