1. INTRODUCTION 5

Theorem 4 follows the spirit of works initiated by Rosenlicht and Singer in [20]

and [28]. These two authors found very close relationships between some continu-

ous derivations and Liouvillian solutions of some polynomial differential equations

(see [3]).

1.3.5. Invariant valuations and singularities of l.d.e. In section 6, our main

result is:

Theorem 5. Let (F/K, ∂) be a Picard-Vessiot extension with differential Ga-

lois group G and ν be a non trivial G-invariant valuation of F/C. Then the fol-

lowing properties hold:

i. If ν is regular for some l.d.e. L = 0 defining F/K, then F/K is solvable.

ii. If furthermore ν = ordt ◦ϕ for some differential embedding ϕ : F → C((t)),

then F/K is algebraic when t = ϕ(t) ∈ K or K = C.

The first point of this result shows that if the center of an invariant valuation

is a regular point of a l.d.e., then the group must be solvable. Therefore, in general,

invariant valuations must be related to the singularities of any l.d.e. that defines

the Picard-Vessiot extension.

1.3.6. Existence and geometry of invariant valuations. In section 7, our main

result is:

Theorem 6. Let F/K be a Picard-Vessiot extension with an algebraically

closed field of constants C, and a connected differential Galois group G of dimension

bigger or equal than one. Then the following hold:

i. There exist non-trivial G-invariant valuations of F/K for which the derivation

is continuous.

ii. Denote by Π the differential algebra generated over K by the elements t ∈

T (F/K)∗, having a pole at some G-invariant place of F/K for which ∂ is

continuous, if G is a simple group, then F coincides with the fraction field of

Π.

The main ideas of this section are the following: the algebraic nature of the

group of automorphism allows or forbids the existence of invariant valuations. For

example, if G is an Elliptic curve i.e. an Abelian variety, Proposition 25(i) asserts

that invariant valuations of F/K cannot exist. However, when G is a connected

aﬃne algebraic group, Theorem 59 asserts that invariant valuations always exist. It

can be viewed as a fixed point theorem for the action of a connected aﬃne group on

the Riemann-Zariski variety. Theorem 6(i) will be a consequence of this fixed point

Theorem. Theorem 6(ii), will be interpreted as a partial converse of Corollary 2.