CHAPTER 1

Introduction

The present work lies at the junction of Differential Galois theory and Valuation

theory. Its mainspring consists in analysing the infinitesimal behaviour of deriva-

tions in abstract differential algebra, with the help of the theory of Valuations.

Before exploring this idea, let’s briefly recall the two concerned theories.

1.1. Differential Galois theory. By an ordinary differential field extension

of characteristic zero (F/K, ∂), we mean a field extension of characteristic zero,

where ∂ is a derivation of F whose restriction to K is a derivation of K. Denote

by CF = Ker ∂ and CK = Ker ∂|K the subfields of constants; When C = CF = CK

is algebraically closed, we say that (F/K, ∂) is without new constants.

Throughout this paper, (F/K, ∂) will denote an ordinary differential exten-

sion of characteristic zero without new constants. Its differential Galois group,

Gal∂(F/K) is the group of K-automorphisms of F commuting with the derivation

∂, i.e.

Gal∂(F/K) := {σ ∈ Aut(F/K)|σ ◦ ∂ = ∂ ◦ σ}.

This group is analogous to the classical Galois group in the sense that it permutes

the solutions of any given polynomial differential equation with coeﬃcients in K.

We will say that an element z ∈ F is holonomic over K and will denote by

T (F/K) the set of all of them, if and only if there is a monic linear differential

equation, (in short l.d.e.) L ∈ K[∂] which annihilates z, i.e. L(z) = 0 where

L =

∂n

+

an−1∂n−1

+ · · · + a1∂ + a0 ∈ K[∂].

The set T (F/K) is a K-algebra and a G = Gal∂(F/K)-module, see Proposition 15

below.

Let (F/K, ∂) and L ∈ K[∂] as above. Thanks to the theory of Wronskian

determinants, one can prove that the set SolF (L = 0) of solutions of L = 0 be-

longing to F , is a C-vector space of dimension bounded by n = ord(L). When

dim(SolF (L = 0)) = n and F is differentially generated by this set of solutions, we

say that (F/K, ∂) is a Picard-Vessiot extension. In this case G = Gal∂(F/K) is

a linear algebraic group over C. Proposition 15 below gathers some classical results

and references about this theory.

A characterisation of conjugated elements under the differential Galois group is

the following: Let f1 and f2 be two elements belonging to a Picard-Vessiot extension

F/K. They are conjugated (i.e there exists σ ∈ Gal∂ (F/K) such that σ(f1) = f2),

iff they are algebraically equivalent, that is: for any polynomial differential equation

P = 0 with coeﬃcients in K

P (f1) = 0 ⇔ P (f2) = 0.

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