Valuations invariantes pour l'action des groupes de Galois différentiels

Abstract

Let ( F / K , ∂ ) be a differential field extension with differential Galois group G = Gal ∂ ( F / K ) . For the natural action of G on the Riemann–Zariski variety S ⋆ = S ⋆ ( F / K ) of the field extension F / K , we study the invariant valuations ν ∈ ( S ⋆ ) G 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. To cite this article: G. Duval, C. R. Acad. Sci. Paris, Ser. I 339 (2004).