2. INVARIANT VALUATIONS AND SOLUTIONS OF L.D.E. 13

Therefore, we get the following inductive property (In) for all n ∈ :

w(n)

= Pn(u) · w,

where Pn(u) is a monic polynomial expression of degree n in u with coeﬃcients

depending on α and its derivatives.

Since w ∈ T (F/K), it satisfies a homogeneous linear differential equation with

coeﬃcients in K:

w(n)

+ an−1 ·

w(n−1)

+ · · · + a1 · w + a0 · w = 0. Hence,

Pn(u) · w + an−1 · Pn−1(u) · w + · · · + a1 · u · w + a0 · w = 0

So,

(8) Pn(u) + an−1 · Pn−1(u) + · · · + a1 · u + a0 = 0.

Equation (8) is a monomial algebraic relation of degree n in u, with coeﬃcients

depending on the ai’s, α, and their derivatives. Since α is algebraic over K, K[α]/K

is a differential field extension. Therefore, all the coeﬃcients of (8) belong to K[α].

Hence, u is algebraic over K, and we have the inequality

(9) [K[u]/K] [K[α]/K] · ordK (w) ∞.

Hence,

w

w

= u is algebraic over K and w is a unit of T (F/K) by Harris-Sibuya’s

theorem (see Proposition 15 (iv)).

(ii) Let I be a nonzero ideal of T (F/K), and let z = 0 belong to I. Let V (z) be

the C-vector space spanned by the conjugates of z under Gal∂(F/K). Then, V (z)

is a finite dimensional vector space by Proposition 15(ii). Let {z = z1,...,zd} be

some of the conjugates of z which form a C-basis of V (z). Let w = W (z1,...,zd) be

the associated Wronskian determinant as in (7). Now assume that I is a differential

ideal, (respectively a Gal∂(F/K)-module). By expanding w by the first column,

(respectively by the first row), one gets in both cases that w ∈ I. To conclude

the proof of (ii), we just have to prove that the condition F

Gal∂ (F/K)

⊂ T (F/K)

implies that w is a unit of T (F/K).

The linear representation ρ : Gal∂(F/K) −→ GlC (V (z)) gives:

σ(w) = det ρ(σ) · w

σ(w ) = det ρ(σ) · w

σ(

w

w

) =

w

w

Hence, u =

w

w

∈ F Gal∂ (F/K), which is a field. So if F Gal∂ (F/K) ⊂ T (F/K), u is a

unit of T (F/K), and so is w by (i) and Proposition 15(i).

Proof of Theorem 1 and Corollary 2.

Let ν be a non-trivial G-invariant valuation of F . If T (F/K) is not contained

in Rν, by definition there exists some t in T

(F/K)∗

such that ν(t) 0, and Case

1 of Theorem 1 holds. Now suppose that T (F/K) ⊂ Rν. Since ν is G-invariant,

mν is a G-stable ideal of Rν. The center P = mν ∩ T (F/K) of ν in T (F/K), is

therefore a prime G-stable ideal of T (F/K). By Proposition 16(ii), it follows that

P is trivial i.e. P = 0 or P = T (F/K).

Since ν is non-trivial, 1 ∈ P , and we can exclude the case P = T (F/K). So we

have P = mν ∩ T (F/K) = {0}. Which gives T

(F/K)∗

⊂ U(Rν) and proves the

theorem.

The proof of the corollary follows from the two statements below: