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 ⊗K T (F/K)
¯
K ⊗C Γ(G(C)) = Γ(G(
¯
K )),
where
¯
K 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:

˜
T =
¯
K ⊗K T (F/K) is an integral domain.
Its field of fractions
˜
F is a Picard-Vessiot extension of
¯
K .
T (
˜
F/
¯
K ) =
˜
T =
¯
K ⊗K T (F/K).
Gal∂(
˜
F/
¯
K ) Gal∂ (F/K) = G(C), where G(C) acts on
˜
T leaving each
element belonging to
¯
K 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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