1. INTRODUCTION 3

Here, we choose to pay attention to valuations which are left invariant by the

action of the differential Galois group Gal∂(F/K). The reasons justifying this se-

lection are the following. On one hand, when two functions f1 and f2 are conjugate

with respect to the differential Galois group, they share the same algebraic and

differential properties. Which means that they both satisfy the same algebraic

and differential equations over the given ground field K. On the other side, val-

uations measure infinitesimal behaviours of functions like orders of vanishing at

some given point or asymptotic growths. For two conjugate functions f1 and f2,

we have ν(f1) = ν(σ(f1)) = ν(f2) if ν is an invariant valuation. Therefore, looking

at invariant valuations means looking at what could be the common infinitesimal

behaviour of two conjugate functions. In other words, our interest for invariant val-

uations is motivated by the following question: If two functions equally behave

from an algebro-differential point of view, do they have any infinitesimal

behaviour in common? And what could be this infinitesimal behaviour?

In order to explore this question properly, we shall focus our attention on the

following two mains topics and interactions between them:

• Invariant valuations, their existence or not, their geometry, their relation

with l.d.e.

• How does a valuation reflect, when it is possible the infinitesimal shape of

functions. Here the notions of continuity of a derivation w.r.t. a valuation

and analytic valued fields will play a central role.

Since the Picard-Vessiot theory is the best known differential Galois theory, most

of the results presented here will only concern this theory. This general philosophy

being set, let us now introduce the plan of the paper.

1.3.1. Invariant valuations and solutions of linear differential equations (l.d.e.)

Our main results in section 2 are

Theorem 1. Let F/K be an ordinary differential field extension with the same

constant field C algebraically closed, and a differential Galois group G such that

F

G

⊂ T (F/K). Let ν be a non trivial G-invariant valuation of F . Then one of the

following statement is true,

• Case 1: There exists t ∈ T

(F/K)∗such that ν(t) 0.

• Case 2: T (F/K)∗ ⊂ U(Rν).

Corollary 2. Let F/K be a Picard-Vessiot field extension and ν a non-trivial

G-invariant valuation of F . Then there exists t ∈ T

(F/K)∗

such that ν(t) 0.

This latter result shows that in the Picard-Vessiot context, for any invariant

place there exists an holonomic element having a pole at this place. In some sense,

this property can be viewed as a sort of ” Riemann-Roch Property ” for solu-

tions of linear differential equations. In section 2.3, Theorem 17 gives two stability

properties of invariant valuations. One shows first that invariant valuations are

strongly invariant, and then that their computation is reduced to a purely algebro-

geometric problem. This observation will be the central idea of section 7.

1.3.2. Examples and use of invariant valuations. In section 3, we illustrate

Theorem 1 and Corollary 2 on some important examples of invariant valuations.

We also analyse the converse of those results in section 3.3, in the context of elliptic