2. INVARIANT VALUATIONS AND SOLUTIONS OF L.D.E. 15

iii. The induced action on S(F/K)∗ is locally-finite and each G◦(C)-invariant val-

uation ν of F/K remains G◦(K)-invariant.

This theorem illustrates two notions of stability. On the one hand, thanks to

Proposition 10, every invariant valuation of F/C remains strongly invariant. On the

other hand, G◦(K) is in general a much bigger group than G◦(C) and invariance for

the small group implies invariance for the bigger one, when the valuation is trivial

on the ground field K. This latter fact is a powerful tool to compute the invariant

valuations of F/K, since it reduces the problem to a purely algebraic statement

in a way that we can completely forget the Picard-Vessiot structure of the field

extension. This idea will be properly explained in section 7 of this paper.

The following is a finiteness property whose proof is left to the reader:

Lemma 18. Let ν be a non-trivial valuation of the field extension F/C and

{z1,...,zd} be d elements of F . Set:

E = ν(

d

i=1

cizi), c = (c1,...,cd) ∈

Cd\{0}}

⊂ Γν ∪ {∞}.

Then card(E) d.

Proof of Theorem 17.

(i) Let z ∈ T (F/K) and d = ordK (z). Let {z = z1,...,zd} be a C-basis of

V (z) = VectC {σ(z)|σ ∈ G(C)} = SolF {Lz(y) = 0} .

Then:

{ν(σ(z))|σ ∈ G(C)} ⊂ E = ν(

d

i=1

cizi), c = (c1,...,cd) ∈

Cd\{0}

.

Therefore Lemma 18 gives us the inequality of (i). Since F/K is a Picard-Vessiot

field extension, F coincides with the field of fractions of T (F/K). So every x ∈ F

∗

can be written as a fraction x =

z1

z2

with z1,z2 in T

(F/K)∗.

By setting

Ei = {ν(σ(zi))|σ ∈ G(C)},i = 1, 2

Δ = {ν(σ(x))|σ ∈ G(C)}.

We get Δ ⊂ E1 − E2 where, E1 − E2 stands for the Minkowski difference of the

two subsets E1 and E2 of Γν. Therefore

card(Δ) card(E1) · card(E2) ordK (z1) · ordK (z2) ∞.

This concludes the proof of (i).

(ii) With the previous notations, for all z ∈ T (F/K) with ordK (z) = d, the

locally finite action of G(C) on T (F/K) can be described by the following relations

(RC ) :σ(z) =

d

i=1

ai(σ)zi, ∀σ ∈ G(C).

Each coeﬃcient ai(σ) belongs to the ring Γ(G(C)) of regular functions on G(C).

Since Γ(G(K)) = K ⊗C Γ(G(C)) the ai are regular functions onG(K). So, if it is