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.