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
• 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
⊂ 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
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