2. INVARIANT VALUATIONS AND SOLUTIONS OF L.D.E. 11 Conversely, G leaves C globally invariant if and only if the f-adic valuation ν0 is left invariant by G. In this case, by Remark 11, ν0 must also be strongly G-invariant since it is a discrete valuation of rank one. By Proposition 13, there exists a one- to-one correspondence between the set of G-invariant valuations ν composed with ν0, and the set of points q C fixed by G. 2.2. A kind of Riemann-Roch property for holonomic functions. The main results of this section are Theorem 1 and Corollary 2. First we introduce the notions necessary for the proofs. Here, (F/K, ∂)is an ordinary differential field extension of characteristic zero, with the same algebraically closed subfield of con- stant C = CF = CK. Let G = Gal∂(F/K) be the Differential Galois Group of the extension. Let’s denote by T (F/K) the set of elements of F which satisfy a non trivial homogeneous l.d.e. with coefficients in K. Definition 14. For each z T (F/K) there exists a unique nonzero monic homogeneous linear differential operator Lz(y) of smallest order with coefficients in K, such that Lz(z) = 0 ([13]). Lz(y) is called the minimal cancellator of z. We denote by ordK(z) the order of Lz(y) as a linear differential operator. Formally speaking the order function plays the role of the degree function in the theory of algebraic extensions. The following proposition gathers classical properties of T (F/K). Proposition 15. Let (F/K, ∂) be an ordinary differential field extension with the same algebraically closed field of constants C. i. T (F/K) is a K-differential subalgebra of F. ii. T (F/K) is mapped onto itself under the natural action of G = Gal∂(F/K). Furthermore the G-action on T (F/K) is C-linearly locally finite. iii. If F/K is a Picard-Vessiot field extension or more generally is a weakly normal extension (i.e. F Gal∂(F/K) = K), then the order function is characterised as follows ∀z T (F/K), ordK(z) = dimC VectC{σ(z)|σ Gal∂(F/K)}. iv. For all z F , z and 1 z both belong to T (F/K) if and only if z z is algebraic over K. v. If F/K is a Picard-Vessiot extension then ¯ ⊗K T (F/K) ¯ ⊗C Γ(G(C)) = Γ(G( ¯ )), where ¯ is an algebraic closure of K and Γ(G(C)) is the ring of regular func- tions on Gwith its structure of affine linear algebraic group defined over C. Furthermore, the isomorphism is G(C)-equivariant for the natural action on T (F/K) and the action by left translation of the regular functions in Γ(G(C)). vi. Moreover, if G(C) = G◦(C) is connected, then: ˜ = ¯ ⊗K T (F/K) is an integral domain. Its field of fractions ˜ is a Picard-Vessiot extension of ¯ . T ( ˜ ¯ ) = ˜ = ¯ ⊗K T (F/K). Gal∂( ˜ ¯ ) Gal∂(F/K) = G(C), where G(C) acts on ˜ leaving each element belonging to ¯ invariant. Proof. (i) is proved in [30]. The result expresses the formal analogy between solutions of monic linear differential equations and algebraic integers.
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