Valuations and differential Galois groups

82 Pages·2011·0.672 MB·English

by  Guillaume Duval

MEMOIRS
of the
American Mathematical Society
Number 998
Valuations and Differential
Galois Groups
Guillaume Duval
July 2011 • Volume 212 • Number 998 (third of 4 numbers) • ISSN 0065-9266
American Mathematical Society
Number 998
Valuations and Differential
Galois Groups
Guillaume Duval
July2011 • Volume212 • Number998(thirdof4numbers) • ISSN0065-9266
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Duval,Guillaume,1971-
ValuationsanddifferentialGaloisgroups/GuillaumeDuval.
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ISBN978-0-8218-4906-4(alk. paper)
1.Differentialequations,Linear. 2.Differentialalgebraicgroups. 3.Galoistheory. I.Title.
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512(cid:2).32—dc22 2011016003
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Contents
Chapter 1. Introduction 1
Chapter 2. Invariant valuations and solutions of l.d.e. 7
Chapter 3. Examples and use of invariant valuations 17
Chapter 4. Continuity of derivations, geometry and examples 23
Chapter 5. Continuity and field extensions 37
Chapter 6. Invariant valuations and singularities of l.d.e. 49
Chapter 7. Existence and geometry of invariant valuations 55
Bibliography 67
v

Abstract
In this paper, valuation theory is used to analyse infinitesimal behaviour of
solutionsoflineardifferentialequations. ForanyPicard-Vessiotextension(F/K,∂)
with differential Galois group G, we look at the valuations of F which are left
invariant by G. The main reason for this is the following: if a given invariant
valuationν measures infinitesimal behaviour of functions belonging toF, then two
conjugateelementsofF willsharethesameinfinitesimal behaviourwithrespectto
ν. The article is divided into seven sections as follows. In section 1, we give a brief
accountonPicard-Vessiottheoryandvaluationtheory. Insection2,weexplorethe
links between invariant valuations and solutions of linear differential equations. To
thisrespect,Corollary2isakindofRiemann-Rochpropertywhichstatesthatsome
solutionsoflinear differential equationsmust havepolesatinvariant valuations. In
section 3, we give examples of invariant valuations. We also use the above theory
to give a new proof of a result due to Drach and Kolchin about elliptic functions
(Theorem 26). In section 4, we analyse the properties of valuations which are
describing the analyticshape of functions. The notionof continuity of a derivation
w.r.t. a valuation, plays a central role. This is justified geometrically thanks to
the theory of vector fields in Corollary 38. In section 5, the permanence of the
continuity by field extensions is proved for algebraic extensions (Theorem 3), and
for invariant valuations of Liouvillian extensions (Theorem 4). In Theorem 5 of
section 6, we show that in general, invariant analytic valuations are related to
singularities of linear differential equations. In section 7, we prove the existence
of invariant valuations (Theorem 6) for Picard-Vessiot extensions with connected
differential Galois group.
ReceivedbytheeditorOctober24,2007and,inrevisedform,April2,2009.
ArticleelectronicallypublishedonDecember3,2010;S0065-9266(2010)00606-9.
2010 MathematicsSubjectClassification. Primary34M15;Secondary12J20.
Key wordsand phrases. DifferentialGaloistheory,valuations,Hardyfields.
Author address at time of publication: 1 Chemin du Chateau, 76 430 Les Trois Pierres,
Franceemail: [email protected].
(cid:2)c2010 American Mathematical Society
vii

CHAPTER 1
Introduction
ThepresentworkliesatthejunctionofDifferentialGaloistheoryandValuation
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 C =Ker∂ and C =Ker∂| the subfields of constants; When C =C =C
F K K F K
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 Sol (L = 0) of solutions of L = 0 be-
F
longing to F, is a C-vector space of dimension bounded by n = ord(L). When
dim(Sol (L=0))=n and F is differentially generated by this set of solutions, we
F
saythat(F/K,∂)isaPicard-Vessiot extension. InthiscaseG=Gal (F/K)is
∂
alinearalgebraicgroupoverC. Proposition15belowgatherssomeclassicalresults
and references about this theory.
AcharacterisationofconjugatedelementsunderthedifferentialGaloisgroupis
thefollowing: Letf andf betwoelementsbelongingtoaPicard-Vessiotextension
1 2
F/K. They are conjugated (i.e there exists σ ∈Gal (F/K) such that σ(f )=f ),
∂ 1 2
ifftheyarealgebraically equivalent,thatis: foranypolynomialdifferentialequation
P =0 with coefficients in K
P(f )=0⇔P(f )=0.
1 2
1