Valuations and differential Galois groups by Guillaume Duval

By Guillaume Duval

During this paper, valuation conception is used to examine infinitesimal behaviour of recommendations of linear differential equations. For any Picard-Vessiot extension $(F / okay, \partial)$ with differential Galois staff $G$, the writer appears to be like on the valuations of $F$ that are left invariant by means of $G$. the most cause of this can be the next: If a given invariant valuation $\nu$ measures infinitesimal behaviour of services belonging to $F$, then conjugate components of $F$ will proportion a similar infinitesimal behaviour with appreciate to $\nu$. This memoir is split into seven sections

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Since ν is strongly G-invariant, ν(z) = ν(z1 ) = · · · = ν(zd ) by Theorem 17. So, as in the proof of Theorem 17, ν(Δ(z)) ν(z). Remark 41. This property has the following geometric interpretation. e. there exists a variety Y of the same type over C such that X Y ⊗C K and the function field of X is F . Each Δ of Lie(G), defines a tangent vector field on Y hence on X. Since X is complete, each invariant valuation ν possesses a center Z ⊂ X. Proposition 40 asserts that this center remains invariant under the vector field Δ (see [24], [7], [1], and [3] for comments).

The following assertions are equivalent 1. The vector field X is tangent to the formal curve γ. 2. t. ν. 3. t. ν. Proof To say that X is tangent to the formal curve γ means that the two vectors γ (t) = are collinear in (19) s x1 (t) xs (t) and X(γ(t)) = P1 ◦ γ(t) , Ps ◦ γ(t) . It also means that exists a nonzero λ(t) ∈ C((t)) such that x1 (t) = λ(t)P1 ◦ γ(t) = λP1∗ xs (t) = λ(t)Ps ◦ γ(t) = λPs∗ . This also means that for the substitution morphism ϕ : F → C((t)), f → f ∗ we have d d ∂f ∂f (ϕ(f )) = (f ∗ ) = (γ(t)) · x1 (t) + · · · + (γ(t)) · xs (t) dt dt ∂x1 ∂xs ∂f ∂f = λ(t) P1 + · · · + Ps ◦ γ(t) ∂x1 ∂xs = λ(t)(∂f )∗ = λ(t)ϕ(∂f ).

Therefore we must have ν(ϕ) 0. Since this holds for all valuations, K ⊂ T (F/K) ⊂ Rν = K. ν∈S ∗ (F/K) This concludes the proof. 4. 6. Invariant valuations and the Lie algebra of G. Let (F/K, ∂) be a Picard-Vessiot extension with constants C and differential Galois group G. t. any G-invariant valuation ν of F/C, which is going to be the main purpose of the next section, we are going to focus on other derivations of F . 27, p. 20), the Lie algebra of G can be identified with the C-vector space of all K-derivations Δ of F commuting with ∂: Lie(G) = {Δ ∈ DerK (F )|[Δ, ∂] = 0}.

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