Valuations and differential Galois groups by Guillaume Duval

By Guillaume Duval

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

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If f = yi with 2 i s, for all p ∈ , we set Æ p fp = yi − an,i y1n . n=1 By equation (17), ϕ(fp ) = n p+1 an,i τ , hence ν(fp ) (p + 1) · d. t. ν so let ω0 ∈ d such that F ∗ ⊂ Lν (ω0 ). We therefore have ν(∂fp ) (p + 1) · d + ω0 . But n·d p ∂fp = ∂yi − an,i ny1n−1 ∂y1 . n=1 Composing by ϕ and computing v = ordτ , we have ordτ (ϕ(∂fp )) = ν(∂fp ), hence, p ordτ (ϕ(∂fp )) = ordτ (ϕ(∂yi ) − an,i nz n−1 ϕ(∂y1 )) (p + 1) · d + ω0 . n=1 Since C((τ )) is complete, by taking the limit when p → +∞, we obtain ∞ an,i nz n−1 ϕ(∂yi ) = ϕ(∂y1 ) n=1 = = d dz ∞ an,i z n ϕ(∂y1 ) n=1 d (ϕ(yi )) ϕ(∂y1 ).

Be the finite set of conjugates of α under Gal(K/K), and let us set fβ .

E. p is a singular point of the vector field. Therefore, the center of invariant valuations belongs to the singularities of any vector field Δ belonging to Lie(G). This observation will be one of our motivation to introduce the geometric development of section 7. However, it seems difficult to relate this property with asymptotic behaviour of functions belonging to F , because, except for constant linear differential equations, there are no explicit relations between these functions and the integral curves of the vector fields Δ.

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