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
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.
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