Liouville integrability: An effective Morales–Ramis–Simó theorem

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Liouville integrability: An effective Morales–Ramis–Simó theorem

A. Aparicio-Monforte, Thomas Dreyfus, Jacques-Arthur Weil

To cite this version:

A. Aparicio-Monforte, Thomas Dreyfus, Jacques-Arthur Weil. Liouville integrability: An effective Morales–Ramis–Simó theorem. Journal of Symbolic Computation, Elsevier, 2016, 74, pp.537-560.

�10.1016/j.jsc.2015.08.009�. �hal-01897311�

MORALES-RAMIS-SIMÓ THEOREM

A. APARICIO-MONFORTE Dawson College,

Westmount, Montreal QC, Canada

T. DREYFUS

Université Paul Sabatier - Institut de Mathématiques de Toulouse, 118 route de Narbonne, 31062 Toulouse, France

J.-A. WEIL

XLIM - Université de Limoges

123 avenue Albert Thomas, 87060 Limoges Cedex, France

Abstract. Consider a complex Hamiltonian system and an integral curve. In this paper, we give an effective and efficient procedure to put the variational equation of any order along the integral curve in reduced form provided that the previous one is in reduced form with an abelian Lie algebra. Thus, we obtain an effective way to check the Morales-Ramis-Simó criterion for testing meromorphic Liouville integrability of Hamiltonian systems.

E-mail addresses: [email protected], [email protected], [email protected].

Date: September 1, 2015.

2010 Mathematics Subject Classification. Primary 37J30, 34A05, 68W30, 34M03, 34M15, 34M25, 17B45.

Secondary 20G45, 32G81, 34M05, 37K10, 17B80 70H06 .

Key words and phrases. Hamiltonian Systems, Ordinary Differential Equations, Complete Integrability, Differential Galois Theory, Computer Algebra, Lie Algebras.

1

Contents

1. Introduction 2

2. The Morales-Ramis-Simó Integrability Condition 5

2.1. Hamiltonian Systems and Liouville Integrability 5

2.2. Variational Equations 6

2.3. Differential Galois Theory and Reduced Forms 7

2.4. The Morales-Ramis-Simó Integrability Criterion. 9

2.5. The Strategy for an effective Morales-Ramis-Simó Criterion. 10 3. Reduction of Linear Differential Systems with a Reduced Abelian Diagonal Part 11

3.1. The Diagonal and Off-Diagonal Subalgebras 11

3.2. The Shape of the Reduction Matrix 12

3.3. The Adjoint Action 14

3.4. Decreasing the Dimension of Lie p A; k q . 16

4. Back to the Morales-Ramis-Simó Integrability Criterion 20

4.1. Reducing the First Variational Equation 20

4.2. The Effective Morales-Ramis-Simó Integrability Criterion 21

4.3. A Simplified Reduction Procedure 22

5. An Example 22

5.1. First Variational Equation 23

5.2. Second Variational Equation 24

5.3. Third Variational Equation 25

6. Conclusion 25

References 26

1. Introduction

Consider a Hamiltonian system of 2n differential equations p X

H

q :

$ &

% 9

q

_i

^B_B^H_pi

9

p

_i ^B_B^H_q

i

A first integral is a function of the q

_i

and p

_i

which is constant along the solutions of p X

_H

q . The system is called (meromorphically) Liouville integrable (or completely integrable ) when it admits n (meromorphic) first integrals F

₁

, . . . , F

_n

which are functionally independent (their differentials are linearly independent) and in involution (their Poisson brackets vanish or, equivalently, the associated Hamiltonian vector fields X

_Fi

commute). We refer to the reference books [AM78, CB97, Aud08] for more on this topic; see also Section 2 for definitions.

The Ziglin-Morales-Ramis theory (see [MRR10, Aud08] for statements and applications)

provides mathematical tools to check when a system is non-integrable. This is particularly

useful as Hamiltonian systems generally come as parametrized families. The non-integrability

criteria allow one to discard the vast majority of values of the parameters for which the

system is not integrable. The principle is as follows. First, we find a particular solution Γ of the system p X

_H

q (generally from an invariant plane found from symmetries) and we compute variational equations pVE

p

q , i.e. systems of linear differential equations governing a Taylor expansion of a solution of p X

_H

q along the particular solution Γ. The Liouville integrability of p X

_H

q induces integrability conditions on the variational equations p VE

_p

q , which in turn imply properties of their monodromy or differential Galois groups. Technically, the Morales-Ramis-Simó theorem states that if p X

_H

q is integrable, then the Lie algebras of the differential Galois groups of all variational equations pVE

p

q must be abelian (all these terms are defined in Section 2).

The strength of this criterion is that it turns a geometric condition (integrability) into an algebraic one (abelianity of a Lie algebra), thus paving the way for possible computations.

However, although there exist general algorithms to compute differential Galois groups of reducible systems such as the variational equations p VE

_p

q ([Fen15, Ret14] or [vdH07]), none of them are currently even close to being practical or implemented at this time.

Furthermore, the size of the variational equations p VE

_p

q grows fast, so only a method which uses the structure of the system to make it simpler has a chance of being efficient. The main goal of the present paper is to explain how to use the structure of the system to make it simpler, which will allow us to check efficiently whether its Lie algebra is abelian or not.

Over the past decade, several approaches have been devised to concretely apply this Morales-Ramis-Simó integrability criterion.

For Hamiltonians of the form H °

_n

i1 1

2

p

²_i

V p q q , where V is a potential in q, the first variational equation is often a direct sum of Lamé equations of the form y

²

p x q p n p n 1 q ℘ p x q B q y p x q , where ℘ denotes the Weierstrass function associated to an elliptic curve. In this case, Morales has elaborated a local criterion to find obstructions to integrability on higher variational equations via local computations (see Lemmas 11 and 12 in [MRR01] Page 79, and Proposition 7, Page 81). Maciejewski, Przybylska and Duval have elaborated techniques to handle variational equations for the case of Hamiltonians with potentials ([MP06, DM09, DM14, DM15]); see also the works of Combot and coauthors [Com13, CK12, BCSED14].

Another approach is to determine numerical trajectories and compute numerical mon- odromies around these. Although it is difficult to obtain rigorous proofs by these methods, they provide surprisingly precise information. They have been developed, for example, by Martinez and Simó [MS09], by Simon and Simó in the Atwood paper [PPR 10], by Simon in the more recent [Sim14a, Sim14b] and by Salnikov [Sal14, Sal13].

The general strategy for turning numerical evidence into rigorous proofs is to show that a certain commutator is non-zero. This in turn yields calculations of integrals and of residues, which can be achieved algorithmically due to their D-finiteness. This is used by Martinez and Simó in [MS09] and later systematized by Combot and coauthors, see e.g.

[CK12, Com13, BCSED14].

The approach that we develop in this paper follows previous work by two of the authors in [AMCW13, AMW11, AMW12]. We establish a reduction method. Consider the p-th variational equation pVE

p

q : Y

¹

ApxqY , where the coefficients of Apxq are in a differential field k. Given an invertible matrix P p x q (a gauge transformation matrix), performing the linear change of variable Z P p x q Y produces an equivalent linear differential system for Z, denoted by Z

¹

P p x qr A p x qs Z. The principle of reduction methods is to look for a gauge transformation P p x q such that the resulting system Z

¹

P p x qr A p x qs Z is “as simplified as possible”.

Let G denote the differential Galois group of p VE

_p

q and g be the Lie algebra of G.

Following traditional works of Kolchin and Kovacic, we will say that we have a reduced form when P p x qr A p x qs P g p k q (see Subsection 2.3.3). Despite the apparent technicality of this definition, the Kolchin-Kovacic theory shows why this is a desirable form. This is similar to the Lie-Vessiot-Guldberg theories of reduction of connections (see [BSMR10, BSMR12] for the latter and their connections with the Kolchin-Kovacic theory of reduced forms). Our strategy in this paper is to compute such a reduction matrix Ppxq efficiently.

After this reduction process, the Lie algebra g is easily read and its abelianity (or not) is given in the process. Furthermore, if g is abelian, then this process will have prepared the system to allow an efficient reduction of the next variational equation.

Our strategy can be summarized as follows. The p-th variational equation p VE

p

q is a differential system of the form Y

¹

A p x q Y where A p x q has the form

A p x q

A

1

pxq 0 S p x q A

2

p x q

.

In the Morales-Ramis-Simó situation (see Subsection 2.5), we may assume that the A

i

pxq are in reduced form and that the Lie algebra of the differential Galois group of the block diagonal system

Y

¹

A

_diag

Y, with A

_diag

A

₁

p x q 0 0 A

₂

p x q

,

has an abelian Lie algebra. We show (Theorem 3.3 in Subsection 3.2) that the reduction matrix may be chosen of the form

P p x q

° Id 0

i

f

_i

p x q S

_i

Id

where Id denotes the identity matrix, where the S

_i

are easily found from S p x q and where the unknown functions f

i

pxq remain to be found. In Subsection 3.4, we show how standard linear algebra allows us to find these f

_i

p x q as rational solutions of first order linear differential equations y

¹

λ p x q y °

i

c

_i

b

_i

p x q where the c

_i

are constant and where λ p x q and the b

i

p x q are in a convenient field.

Structure of the paper. In Section 2, we recall the necessary notions of Liouville

integrability of Hamiltonian systems, differential Galois groups, reduced forms of linear

differential systems and the Morales-Ramis-Simó integrability condition. This section

contains only previously known material. In Section 3, we solve a problem that is interesting in its own right : given a block triangular differential system whose diagonal blocks are in reduced form and have an abelian Lie algebra, we give a practical procedure to put the system into reduced form (and hence compute its differential Galois group). In Section 4, we show how to reduce the Morales-Ramis-Simó condition to the latter problem and thereby provide an effective version of the Morales-Ramis-Simó integrability criterion. In Section 5, we demonstrate the efficiency of the method by computing the reduced form and the differential Galois groups of the first three variational equations on a four dimensional example, originally considered in [CDMP10].

Acknowledgments. We would like to thank G. Casale, T. Combot, A. Maciejewski, J.-J. Morales, M. Przybylska, J.-P. Ramis and M.F. Singer for inspiring conversations regarding the material elaborated here. This paper was initiated at an EMS conference in Bedlewo and significantly improved in Wuhan where T. Dreyfus and J.-A. Weil were invited by the Chinese Academy of Science and the ANR project q-diff. T. Dreyfus is supported by the Labex CIMI in Toulouse.

2. The Morales-Ramis-Simó Integrability Condition

2.1. Hamiltonian Systems and Liouville Integrability. Let p M , ω q be a complex analytic symplectic manifold of complex dimension 2n with n P N

. Since M is locally isomorphic to an open connected domain U € C

²ⁿ

, Darboux’s theorem allows us to choose a set of local coordinates p q , p q p q

₁

. . . q

_n

, p

₁

. . . p

_n

q in which the symplectic form ω is expressed as J :

0 Id_n

Id_n 0

, where Id

n

denotes the identity matrix of size n. In these coordinates, given a function H P C

²

pU q : U ÝÑ C (the Hamiltonian), we define a Hamiltonian system over U € C

²ⁿ

as the differential equation given by the vector field

X

_H

: J∇H

¸

n i1

B H Bp

i

B Bq

i

¸

n i1

B H Bq

i

B Bp

i

, corresponding to the Hamiltonian differential system

(2.1) q 9

_i

^B_B^H_pi

p q , p q , p 9

_i ^B_B^H_q

i

p q , p q , for i 1 . . . n.

Consider a non-punctual integral curve Γ of (2.1). A meromorphic function F : U ÝÑ C is called a meromorphic first integral of (2.1) along Γ if it is constant along integral curves in a neighborhood of Γ, or equivalently when X

_H

p F q 0. Observe that the Hamiltonian is a first integral of (2.1), as we clearly have X

H

pHq 0.

The Poisson bracket t , u of two meromorphic functions f, g P C

²

p U q is de- fined by t f , g u : x ∇f , J∇g y . In the Darboux coordinates, its expression is t f , g u

¸

n i1

B f B q

i

B g B p

i

B f

B p

i

B g B q

i

. The Poisson bracket endows the set of first integrals with a

structure of Lie algebra. A function F is a first integral of (2.1) if and only if t F , H u 0,

i.e. H and F are in involution. Also, note that X

_{tF , Hu}

r X

_F

, X

_H

s , so the involution

condition means that the associated Hamiltonian vector fields commute.

A Hamiltonian system with n degrees of freedom is called Liouville integrable by mero- morphic first integrals along the integral curve Γ if it possesses n first integrals (including the Hamiltonian) meromorphic over U which are functionally independent and in pairwise involution.

2.2. Variational Equations. Among the various approaches to the study of meromorphic integrability of complex Hamiltonian systems, we choose a Ziglin-Morales-Ramis type of approach. Concretely, our starting points are the Morales-Ramis theorem [MRR01] and its generalization, the Morales-Ramis-Simó theorem [MRRS07, MRR10]. These two results give necessary conditions for the meromorphic integrability of Hamiltonian systems. Here, we need to introduce the notion of variational equation of order p P N

along a non-punctual integral curve of (2.1).

Let Φpz, tq be the flow defined by the equation (2.1). Given a non-punctual integral curve Γ of (2.1) and z

₀

P Γ, we let φ p t q : Φ p z

₀

, t q denote a temporal parametrization of Γ. We define the p

^th

variational equation p VE

^p_φ

q of (2.1) along Γ to be the differential equation satisfied by the ξ

_j

:

^BjΦ_Bz^{pz , t}j ^q

for j ¤ p. For instance, the first three variational equations are given by (see [MRRS07], §3.4, Equation p 14 q , Page 860):

pVE

³_φ

q :

$ '

&

' %

p VE

²_φ

q :

# p VE

¹_φ

q : ξ 9

1

d

_φ

X

H

ξ

1

ξ 9

₂

d

²_φ

X

_H

p ξ

₁

, ξ

₁

q d

_φ

X

_H

ξ

₂

ξ 9

₃

d

³_φ

X

_H

p ξ

₁

, ξ

₁

, ξ

₁

q 2d

²_φ

X

_H

p ξ

₁

, ξ

₂

q d

_φ

X

_H

ξ

₃

. For p 1, the first variational equation pVE

¹_φ

q is a linear differential equation

ξ 9

₁

A

₁

ξ

₁

where A

₁

: d

_φ

X

_H

J Hess

_φ

p H q P sp p n , Cx φ p t qyq ,

where Cx φ p t qy denotes the differential field generated by the coefficients of the parametriza- tion φptq . Higher order variational equations are not linear for p ¥ 2. However, for every p VE

^p_φ

q , one can construct an equivalent linear differential system p LVE

^p_φ

q called the linearized p

^th

variational equation (see [MRRS07], §3.4 and [Sim14b]). Indeed, p VE

^p_φ

q is linear in ξ

_p

and polynomial in the ξ

i

for i   p; however, the ξ

i

for i   p are solutions of the linear differential system p LVE

^p_φ¹

q so that polynomials in the ξ

_i

also satisfy linear differential systems, obtained via symmetric powers and tensor constructions. See, for example, §3 of [AMCW13] for practical details on these tensor constructions on differential systems.

For example, p VE

²_φ

q is linear in ξ

2

and linear in the monomials of degree 2 in the ξ

1

, i.e. in the solutions of the second symmetric power system Y

¹

sym

²

p A

1

q Y . Hence the system pLVE

²_φ

q is lower block-triangular and its diagonal blocks are sym

²

pA

1

q and A

1

. We obtain (see e.g. [MRR10, AMW11, Sim14b, CW15]) the following matrices A

p

for the first p LVE

^p_φ

q :

A

₂

p x q

sym

²

p A

₁

p x qq 0 S

₂

p x q A

₁

p x q

,

A

3

p x q

sym

³

pA

1

pxqq 0 S

3

pxq A

2

pxq

sym

³

pA

1

pxqq 0 0 S

3,2

pxq sym

²

pA

1

pxqq 0 S

3,1

p x q S

2

p x q A

1

p x q

.

In general, the matrix of pLVE

^p_φ

q is of the form A

_p

p x q

sym

^p

p A

₁

p x qq 0 S

_p

p x q A

_p1

p x q

.

In [Sim14b], §4.1, Simon provides explicit formulas for these linearized variational equations.

In what follows, we will identify p VE

^p_φ

q and p LVE

^p_φ

q and we will just speak of variational equations of order p.

The matrix sym

ⁱ

pA

1

pxqq has

^{n i}_n1¹

rows and columns, so that pLVE

^p_φ

q is a first order linear differential system of d

p

: °

_p

i1 n i1

n1

^{n p}_n

1 equations. The size d

p

grows fairly fast (polynomially of degree n in p) and forbids the use of a generic algorithm to compute on p LVE

^p_φ

q . For this reason, we elaborate a specific algorithm which takes advantage of the structure of p LVE

^p_φ

q so that the polynomial growth of the size will become a relatively minor concern.

2.3. Differential Galois Theory and Reduced Forms. We begin this subsection with elements of differential Galois theory. We refer to [PS03] or [CH11, Sin09] for details and proofs.

2.3.1. The Base Field. Our base field will be k : C xφptqy , the differential field generated by the coefficients of the parametrization φptq (and C is the field of constants, which is assumed to be algebraically closed). We need to make assumptions about k to elaborate our algorithms. First we assume that k is an effective field, i.e. that one can compute representatives of the four operations , , , { and one can effectively test whether two elements of k are equal, see e.g. [Sin91]. Secondly, we assume that, given any scalar linear differential equation L p y p x qq 0 where

L p y p x qq : a

n

p x q y

^pnq

p x q a

n1

p x q y

^pn1q

p x q a

1

p x q y

¹

p x q a

0

p x q y p x q , with a

i

p x q P k, one can effectively compute a basis of its space of rational solutions, i.e. the solutions which are in the base field k. The standard example of such a field would be k C p x q with C Q . Singer showed, in [Sin91], Lemma 3.5, that if k is an elementary extension of C pxq or if k is an algebraic extension of a purely transcendental Liouvillian extension of C pxq , then k satisfies the above two conditions and hence suits our purposes. He also proved, see Theorem 4.1 in[Sin91], that an algebraic extension of k still satisfies our two assumptions, which will be useful, as reducing the first variational equation may induce algebraic extensions.

2.3.2. Differential Galois Theory. Let us consider a linear differential system of the form

Y

¹

p x q A p x q Y p x q , with A p x q P M

_n

p k q , that is a square matrix of size n P N

in coefficients

in k. A Picard-Vessiot extension for Y

¹

p x q A p x q Y p x q is a differential field extension K | k,

generated over k by the entries of a fundamental solution matrix and such that the field of

constants of K is C. The Picard-Vessiot extension K exists and is unique up to differential

field isomorphism.

The differential Galois group G of Y

¹

pxq ApxqY pxq is the group of field automorphisms of the Picard-Vessiot extension K that commute with the derivation and leave all elements of k invariant. Let U pxq P GL

n

pKq be a fundamental solution matrix of Y

¹

pxq ApxqY pxq with coefficients in K. For any ϕ P G, ϕ p U p x qq is also a fundamental solution matrix, so there exists a constant matrix C

ϕ

P GL

n

C

such that ϕ p U p x qq U p x q .C

ϕ

. The map ρ

_U

: ϕ ÞÝÑ C

_ϕ

is an injective group morphism. An important fact is that G, identified with Im ρ

_U

, may be viewed as a linear algebraic subgroup of GL

_n

C

.

Two linear differential equations Y

¹

p x q A p x q Y p x q and Y

¹

p x q B p x q Y p x q , with A p x q , B p x q P M

_n

p k q are said to be equivalent over k (or gauge equivalent over k) when there exists P p x q P GL

n

p k q , called a gauge transformation matrix, such that

B p x q P p x q r A p x qs : P p x q A p x q P

¹

p x q P

¹

p x q P

¹

p x q . Note that in this case:

Y

¹

p x q A p x q Y p x q ðñ r P p x q Y p x qs

¹

B p x q P p x q Y p x q .

Conversely, if there exist matrices A p x q , B p x q P M

_n

p k q and P p x q P GL

_n

p k q , such that we have Y

¹

pxq ApxqY pxq , Z

¹

pxq B pxqZpxq and Z pxq P pxqY pxq , then

B p x q P p x q r A p x qs .

The Lie algebra g of the linear algebraic group G € GL

_n

C

is the tangent space to G at the identity. Equivalently, it is the set of matrices N such that Id

n

N satisfies the defining equations of the algebraic group G modulo

²

.

Part two of the following proposition is known as the Kolchin-Kovacic reduction theorem.

A proof can be found in [PS03], Proposition 1.31 and Corollary 1.32. See also [BSMR10], Theorem 5.8.

Proposition 2.1 (Kolchin-Kovacic reduction theorem). Let us consider the differential system Y

¹

p x q A p x q Y p x q with A p x q P M

_n

p k q . Let G be its differential Galois group and g be the Lie algebra of G.

(1) Let H € GL

_n

C

be a linear algebraic group and h € M

_n

C

be its Lie algebra. If A p x q belongs to h p k q : h b

_C

k, then G is contained in a conjugate of H.

(2) Assume that k is a C

¹

-field

and G is connected. Let H  G be a connected linear algebraic group with Lie algebra h such that A p x q P h p k q . Then, there exists a gauge transformation P p x q P H p k q such that P p x qr A p x qs P g p k q .

2.3.3. Reduced Forms of Linear Differential Systems. Let Apxq P M

_n

pkq , G be the differen- tial Galois group of Y

¹

p x q A p x q Y p x q and g its Lie algebra.

We say that the system Y

¹

p x q A p x q Y p x q is in reduced form (or in Kolchin-Kovacic reduced form) when A p x q P g p k q g b

C

k. This section contains a quick survey on reduced forms and their practical use.

A field k is a C

¹

-field when every non-constant homogeneous polynomial P over k has a non-trivial zero

provided that the number of its variables is more than its degree. For example, C p x q is a C

¹

-field and any

algebraic extension of a C

¹

-field is a C

¹

-field.

Following [WN63], a Wei-Norman decomposition of Apxq is a finite sum of the form A p x q ¸

a

i

p x q M

i

,

where M

_i

has coefficients in C and the a

_i

p x q P k form a basis of the C-vector space spanned by the entries of A p x q . The M

_i

depend on the choice of a

_i

p x q but the C-vector space generated by the M

i

is independent of the choice of the a

i

pxq .

Definition 2.2. Let Lie p A q € M

_n

C

denote the Lie algebra generated by the M

_i

. We define Lie

alg

p A q € M

_n

C

, called the Lie algebra associated to A, as the algebraic envelope of the Lie algebra Lie p A q , i.e. as the smallest Lie algebra of a linear algebraic group which contains LiepAq .

Let LiepA; kq : LiepAqpkq € M

_n

pkq and Lie

alg

pA; kq : Lie

alg

pAqpkq € M

_n

pkq . We see that the system Y

¹

p x q A p x q Y p x q is in reduced form when Lie

_alg

p A; k q g p k q .

These reduced forms have long been studied in the context of inverse problems in differential Galois theory (see [MS02] and references therein). Their use in direct problems is more recent. Blazquez and Morales use them in their studies of Lie-Vessiot systems in [BSMR10, BSMR12]. Their application to Morales-Ramis theory is initiated in [AMW12]

where Aparicio-Monforte and Weil show how to put the first variational equation in reduced form. In [AMCW13], the same authors with Compoint show that a system is in reduced form if and only if, for any tensor construction constp A p x qq on A p x q , any rational or hyperexponential solution of Y

¹

const p A p x qq Y has constant coefficients. One can also find in [AMCW13] a complete procedure to put a linear differential system into reduced form when it is irreducible (or completely reducible). This does not really apply here as the variational equations are generally reducible (and not completely reducible) systems – and the reduction method of [AMCW13] is far from being efficient yet.

The approach that we elaborate in this paper was initiated (incompletely) in [AMW11].

It is based on another criterion for reduced form, which is given in the following lemma.

Lemma 2.3. Given A p x q P M

_n

p k q , let G be the differential Galois group of Y

¹

p x q A p x q Y p x q and g be its Lie algebra. Let H be a connected linear algebraic group whose Lie algebra h satisfies h Lie

alg

pAq . Assume that G is connected.

Then Y

¹

p x q A p x q Y p x q is in reduced form, i.e. G H and g h, if and only if, for all gauge transformation matrices P p x q in H p k q , we have h p k q Lie

_alg

p P r A s ; k q .

Proof. Follows directly from the Kolchin-Kovacic reduction theorem, see Proposition 2.1.

2.4. The Morales-Ramis-Simó Integrability Criterion. We are now in position to state the Morales-Ramis-Simó integrability criterion. See [MRRS07] for a proof and §2 for the definitions.

Theorem 2.4 (Morales-Ramis-Simó integrability criterion). Consider a Hamiltonian vector

field X

_H

and a non-punctual integral curve Γ. For p P N

, let G

_p

be the differential Galois

group of p VE

^p_φ

q , the p

^th

variational equation along Γ. Let g

_p

be the Lie algebra of G

p

.

Assume that the Hamiltonian vector field X

H

is Liouville integrable by meromorphic first integrals along the integral curve Γ. Then, for all p P N

, g

_p

is abelian.

Of course, given p P N

, computing the differential Galois group G

_p

of such a big differential system would be an unrealistic task in practice unless we use the structure of the system to simplify the computations. We will establish a specific reduction method, i.e.

compute a gauge transformation matrix P

_p

p x q such that P

_p

p x qr A

_p

p x qs P g

_p

p k q . After this reduction process, the Lie algebra g

_p

is easily read and its abelianity (or not) is given in the process. Furthermore, if g

_p

is abelian, then this process will have prepared the system to allow an efficient reduction of the next variational equation.

2.5. The Strategy for an effective Morales-Ramis-Simó Criterion. We refer to §2.2 and §2.3 for the notations used in this subsection. Let us fix an integer p ¥ 2. The matrix of the p

^th

variational equation has the form

A

p

pxq

sym

^p

p A

₁

p x qq 0 S

_p

p x q A

_p1

p x q

.

For each m P t 1, . . . , p u , we let G

_m

denote the differential Galois group of the m

^th

variational equation Y

¹

pxq A

m

pxqY pxq and g

_m

its Lie algebra. For all m P t1, . . . , p 1u , we assume that we know a gauge transformation matrix P

_m

p x q such that P

_m

p x qr A

_m

p x qs is in reduced form, i.e. Lie

alg

pP

m

rA

m

sq g

_m

, and we further assume that each g

_m

is abelian. We let A

_m,red

p x q denote the obtained reduced form, that is A

_m,red

p x q : P

m

p x qr A

m

p x qs .

Under these hypotheses, we will show in the next section how to put the p

^th

variational equation A

_p

p x q into reduced form in an efficient way.

Remark 2.5. Our assumptions imply that the first variational equation is in reduced form.

This in turn implies that our base field k is no longer just C xφy but may be an algebraic extension of the latter (see [AMCW13]). In the sequel, our base field k is the algebraic extension of C x φ y which is needed to put the first variational equation into reduced form.

Since an algebraic extension of a C

¹

-field is a C

¹

-field, we obtain that k is a C

¹

-field provided that C x φ y is a C

¹

-field. Consequently, we are allowed to use Proposition 2.1 as soon as C x φ y is a C

¹

-field. From now on, we assume that k is a C

¹

-field.

Our assumptions also imply (see [AMCW13], Lemma 32, Page 1513) that, for all m P t 1, . . . , p 1 u , the differential Galois groups G

m

are connected. Moreover, both the groups G

_m

and their Lie algebras g

_m

are abelian.

Lemma 2.6. The group G

p

is connected.

Proof. This is a direct application of [MRR10], Lemma 10.

As we can see in [AMCW13], Lemma 14, Page 1508,

Sym

^p

p P

₁

p x qqrsym

^p

p A

₁

p x qqs sym

^p

p A

_1,red

p x qq .

Also, sym

^p

p A

_1,red

p x qq is a reduced form of sym

^p

p A

₁

p x qq . Indeed, this follows from [AMCW13],

Theorem 1, because any tensor construction on sym

^p

pA

1

pxqq is a construction on A

1

pxq .

Consider the block-diagonal gauge transformation matrix Q p x q :

Sym

^p

p P

₁

p x qq 0 0 P

_p1

p x q

.

Thanks to the above remarks (see also [AM10], §4.5.2), we find that Q p x qr A

_p

p x qs

sym

^p

p A

_1,red

p x qq 0 S p x q A

_p1,red

p x q

,

where S p x q has entries in k, and the block-diagonal part of Q p x qr A

p

p x qs is in reduced form.

Furthermore, Lie

_alg

sym

^p

p A

1,red

q 0

0 A

p1,red

is abelian.

3. Reduction of Linear Differential Systems with a Reduced Abelian Diagonal Part

The previous subsection shows that finding a reduced form for the p

^th

variational equation now amounts to finding a reduced form for

Apxq : QpxqrA

p

pxqs

sym

^p

p A

_1,red

p x qq 0 S p x q A

_p1,red

p x q

P M

_n

pkq.

The submatrices sym

^p

p A

_1,red

p x qq and A

_p1,red

p x q belong respectively to M

_n1

p k q and M

_n2

pkq , with n

1

:

^{n p}_n1¹

and n

2

:

^{n p}_n¹

1.

We have A p x q A

diag

p x q A

sub

p x q , where A

diag

p x q :

sym

^p

pA

1,red

pxqq 0

0 A

p1,red

pxq

and A

sub

p x q :

0 0

Spxq 0

. We have seen that Y

¹

p x q A

diag

p x q Y p x q is in reduced form and Lie

_alg

p A

_diag

q is abelian. The aim of this section is to show how to use those hypotheses to put the full system Y

¹

pxq ApxqY pxq in reduced form.

3.1. The Diagonal and Off-Diagonal Subalgebras. We refer to §2.3.3 for the no- tations used in this subsection. Let M

1

, . . . , M

δ

P M

_n

C

be a basis of Lie

alg

p A

diag

q and let B

₁

, . . . , B

_σ

P M

_n

C

be a basis of Lie

_alg

p A

_sub

q . We define the vector space h : Lie

_alg

p A

_diag

q ` Lie

_alg

p A

_sub

q . Note that Lie

_alg

p A q „ h, and h is the Lie algebra of a linear algebraic group. Let us sum up some elementary properties of h in the two following lemmas:

Lemma 3.1. Let us consider a matrix

N

₁

p x q 0 N

_2,1

p x q N

₂

p x q

P h p k q and matrices

0 0

C

₁

p x q 0

,

0 0

C

₂

p x q 0

P Lie

_alg

p A

_sub

; k q . (1) For p i, j q P t 1; 2 u

²

,

0 0

C

i

p x q 0

0 0

C

j

p x q 0

0.

(2) The matrix

N

₁

p x q 0 N

_2,1

p x q N

₂

p x q

0 0

C

₁

p x q 0

and the Lie bracket N

₁

p x q 0

N

_2,1

p x q N

₂

p x q

,

0 0

C

₁

p x q 0

belong to Lie

_alg

p A

_sub

; k q . Furthermore Lie

_alg

p A

_sub

; k q is an ideal in h p k q .

Proof. (1) A straightforward computation shows the first point of the lemma.

(2) We have

N

₁

p x q 0 N

_2,1

p x q N

₂

p x q

0 0

C

₁

p x q 0

0 0

N

₂

p x q C

₁

p x q 0

P h pkq and N

₁

p x q 0

N

_2,1

p x q N

₂

p x q

,

0 0

C

₁

p x q 0

0 0

N

₂

p x q C

₁

p x q C

₁

p x q N

₁

p x q 0

P h pkq.

We prove that they belong to Lie

_alg

p A

_sub

; k q using that fact that the diagonal blocks of the two matrices are zero. The latter Lie bracket identity also shows that Lie

_alg

p A

_sub

; k q is an ideal in h p k q .

Lemma 3.2. For all B p x q P Lie

_alg

p A

_sub

; k q , we have exp p B p x qq Id

n

B p x q and log p Id

_n

B p x qq B p x q . This induces two bijective maps which are inverses of each other

exp : Lie

_alg

p A

_sub

; k q ÝÑ !

Id

n

B p x q , B p x q P Lie

_alg

p A

_sub

; k q )

B p x q ÞÑ Id

_n

B p x q

log :

!

Id

n

B p x q , B p x q P Lie

alg

p A

sub

; k q )

ÝÑ Lie

alg

p A

sub

; k q

Id

_n

B p x q ÞÑ B p x q .

Proof. Let B p x q P Lie

_alg

p A

_sub

; k q . The equality exp p B p x qq Id

_n

B p x q is a di- rect consequence of the first point of Lemma 3.1. The same argument shows that log p Id

_n

B p x qq B p x q . It follows directly that exp and log are bijective on the wished

sets and inverses of each other.

3.2. The Shape of the Reduction Matrix. We refer to §2.3 and §3.1 for the notations and definitions used in this subsection. The aim of this subsection is to prove:

Theorem 3.3. There exists a gauge transformation P p x q P !

Id

_n

B p x q , B p x q P Lie

_alg

p A

_sub

; k q ) , such that Y

¹

p x q P p x qr A p x qs Y p x q is in reduced form.

Let G be the differential Galois group of Y

¹

p x q A p x q Y p x q . By construction, we have G G

p

, where G

p

is the differential Galois group of the p-th variational equation Y

¹

p x q A

_p

p x q Y p x q . Let H be the connected linear algebraic group with Lie algebra h.

Before proving Theorem 3.3, we start with a key lemma.

Lemma 3.4. There exists a unipotent gauge transformation Ppxq , of the form P pxq Id

n1

0

N p x q Id

n2

P H p k q , such that Y

¹

p x q P p x qr A p x qs Y p x q is in reduced form.

Proof. Let H

A

be the connected linear algebraic group with Lie algebra Lie

alg

pA; kq . We have the inclusions G „ H

_A

„ H. As G G

_p

, Lemma 2.6 shows that G is connected. So we may use the second point of Proposition 2.1 to obtain the exis- tence of Q r p x q :

D

1

p x q 0 S

Q

p x q D

2

p x q

P H

_A

p k q such that the linear differential system Y

¹

p x q r Q p x qr A p x qs Y p x q is in reduced form. Let R p x q :

D

₁¹

pxq 0 0 D

₂¹

pxq

P H p k q so that R p x q r Q p x q

Id

n1

0

D

₂¹

p x q S

_Q

p x q Id

_n2

P H p k q . Consequently, to prove the lemma, it is sufficient to prove that Y

¹

p x q R p x q r Q p x qr A p x qs Y p x q is in reduced form. We have to prove that Lie

_alg

Q r r A s ; k Lie

_alg

R Q r r A s ; k . Let H

_RQr

be the algebraic group whose Lie algebra is Lie

_alg

R Q r r A s . Thanks to the first point of Proposition 2.1, the group H

_RQr

contains G. Since Y

¹

p x q r Q p x qr A p x qs Y p x q is in reduced form, G is an algebraic group whose Lie algebra is Lie

alg

QrAs r . This implies that Lie

alg

QrAs; r k „ Lie

alg

R QrAs; r k . Let K | k denote the Picard-Vessiot extension for the equation Y

¹

p x q A p x q Y p x q and let U p x q :

U

1

pxq 0 U

2,1

pxq U

2

pxq

P GL

n

p K q , with U

i

p x q P GL

ni

p K q be a fundamen- tal solution. The elements of G are of the form

G

1

0 G

2,1

G

2

P GL

n

C , with G

i

P GL

ni

C

. Let G

sub

be the subgroup of elements of G of the form

Id

n1

0 G

2,1

Id

n2

. A direct computation shows that G

_sub

is a normal subgroup of G. Therefore, G G

sub

o G{G

sub

. Due to [PS03], Proposition 1.34, (2), G

diag

: G{G

sub

is isomorphic to the differential Galois group of Y

¹

p x q A

_diag

p x q Y p x q . Let us write Q r p x qr A p x qs : D

₁

p x qr sym

^p

p A

_1,red

p x qqs 0

A

_2,1

p x q D

₂

p x qr A

_p1,red

p x qs

, for some matrix A

_2,1

pxq in coefficients in k. We use the relation G G

_sub

o G

_diag

and the fact that Y

¹

p x q r Q p x qr A p x qs Y p x q is in reduced form to find that

Lie

_alg

Q r r A s ; k Lie

_alg

D

1

rsym

^p

pA

1,red

qs 0 0 D

2

rA

p1,red

s

p k q ` Lie

_alg

0 0

A

_2,1

0

p k q .

A direct computation shows that (3.1) Rpxq r QpxqrApxqs

sym

^p

pA

1,red

pxqq 0

D

₂¹

pxqA

_2,1

pxqD

1

pxq A

p1,red

pxq

.

By construction,

Lie

alg

R Q r r A s ; k „ Lie

alg

sym

^p

p A

_1,red

q 0

0 A

p1,red

p k q ` Lie

alg

0 0

D

₂¹

A

_2,1

D

1

0

p k q . Since D

1

pxq and D

2

pxq are invertible matrices, Lie

alg

0 0

A

_2,1

0

pkq and Lie

alg

0 0

D

₂¹

A

_2,1

D

1

0

pkq have the same dimension. Due to the inclusion Lie

_alg

Q r r A s ; k „ Lie

_alg

R Q r r A s ; k we obtain that

(3.2) Lie

_alg

0 0

A

_2,1

0

p k q Lie

_alg

0 0

D

₂¹

A

_2,1

D

1

0

p k q .

Using the facts that the systems Y

¹

p x q A

_diag

p x q Y p x q and Y

¹

p x q r Q p x qr A p x qs Y p x q are in reduced form and G G

_sub

o G

_diag

, we find that

Lie

alg

sym

^p

p A

_1,red

q 0

0 A

_p1,red

pkq Lie

alg

D

₁

rsym

^p

p A

_1,red

qs 0 0 D

₂

r A

_p1,red

s

pkq.

Combined with (3.2), this proves that Lie

alg

R QrAs; r k „ Lie

alg

QrAs; r k . Since we have an inclusion Lie

_alg

Q r r A s ; k „ Lie

_alg

R Q r r A s ; k , we obtain the equality Lie

_alg

R Q r r A s ; k Lie

_alg

Q r r A s ; k . In other words, Y

¹

p x q R p x q r Q p x qr A p x qs Y p x q is

in reduced form.

Proof of Theorem 3.3. It follows from Lemma 3.4 that a reduction matrix can always be chosen of the form P p x q

Id

n1

0 N p x q Id

n2

P H p k q , where N p x q P M

_n2,n1

p k q . By a straightforward computation, we find log p P p x qq

0 0

N p x q 0

P h p k q . But with the same reasoning as in the proof of Lemma 3.1, we obtain that log p P p x qq P Lie

_alg

p A

_sub

; k q . This

concludes the proof of Theorem 3.3.

The following corollary will be crucial for the reduction procedure of §3.4.

Corollary 3.5. Assume that, for all gauge transformations of the form P p x q P !

Id

n

B p x q , B p x q P Lie

_alg

p A

_sub

; k q )

, we have Lie p A; k q Lie p P r A s ; k q . Then, Y

¹

pxq ApxqY pxq is in reduced form.

Proof. Theorem 3.3 provides a Bpxq P Lie

alg

pA

sub

; kq and P pxq Id

n

Bpxq such that the system Y

¹

p x q P p x qr A p x qs Y p x q is in reduced form. In virtue of the hypothesis, Lie p A; k q Lie p P r A s ; k q . This implies that Lie

_alg

p A; k q g p k q , where g is the Lie algebra of the differential Galois group G of Y

¹

p x q A p x q Y p x q . This proves that Y

¹

p x q A p x q Y p x q

is in reduced form.

3.3. The Adjoint Action. We refer to §2.3 and §3.1 for the notations and definitions used in this subsection. In §3.2, we have proved the existence of a gauge transformation matrix Ppxq P !

Id

n

Bpxq, Bpxq P Lie

alg

pA

sub

; kq )

, such that Y

¹

pxq PpxqrApxqsY pxq is in reduced form. Let B

₁

, . . . , B

_σ

P M

_n

C

denote a basis of Lie

_alg

p A

_sub

q . Proposition 3.6. If P p x q : Id

_n

¸

σ i1

f

_i

p x q B

_i

, with f

_i

p x q P k and B

_i

P Lie

_alg

p A

_sub

q , then

P pxqrApxqs Apxq

¸

σ i1

f

i

pxqrB

i

, A

diag

pxqs

¸

σ i1

f

_i¹

pxqB

i

.

Proof. Due to the first point of Lemma 3.1, we have the equalities P

¹

pxq Id

n

¸

σ i1

f

i

pxqB

i

and P p x q A p x q A p x q

¸

σ i1

f

_i

p x q B

_i

A

_diag

p x q . As A p x q A

_diag

p x q A

_sub

p x q , we use Lemma 3.1 and find that

P pxqApxqP

¹

pxq

A

diag

pxq A

sub

pxq

¸

σ j1

f

j

pxqB

j

A

diag

pxq Id

n

¸

σ k1

f

k

pxqB

k

A p x q

¸

σ j1

f

j

p x q B

j

A

diag

p x q

¸

σ k1

f

k

p x q A

diag

p x q B

k

A p x q

¸

σ i1

f

i

p x qr B

i

, A

_diag

p x qs .

Similarly, we have

P

¹

p x q P

¹

p x q

_σ

¸

i1

f

_i¹

p x q B

_i

Id

_n

¸

σ j1

f

_j

p x q B

_j

¸

σ i1

f

_i¹

p x q B

_i

.

This yields the desired result.

We have seen in Lemma 3.1 that Lie

_alg

p A

_sub

; k q is an ideal in h p k q . In particular, for all B p x q P Lie

_alg

p A

_sub

; k q , the bracket r B p x q , A

_diag

p x qs is in Lie

_alg

p A

_sub

; k q . This implies that the k-linear map Ψ : r , A

diag

p x qs , which is the adjoint action of Lie

alg

p A

diag

; k q on Lie

_alg

p A

_sub

; k q , is well defined:

Ψ : Lie

alg

pA

sub

; kq ÝÑ Lie

alg

pA

sub

; kq B p x q ÞÝÑ r B p x q , A

_diag

p x qs .

The following lemma will be necessary in §3.4. Note that the proof of the lemma gives a

complete description of a finite set containing the eigenvalues of Ψ.

Lemma 3.7. The eigenvalues of the linear map Ψ belong to k.

Furthermore, there exists a basis of constant matrices, such that the matrix of the linear map Ψ in this basis is block-diagonal, with blocks that are upper-triangular matrices with only one eigenvalue.

Proof. Let M

₁

, . . . , M

_δ

P M

_n

C

be a basis of Lie

_alg

p A

_diag

q , which is abelian. We may write A

_diag

p x q

¸

δ i1

g

_i

p x q M

_i

with g

_i

p x q P k. Let Ψ

_i

: r , M

_i

s denote the adjoint action of M

_i

on Lie

_alg

p A

_sub

q . As the matrices M

_i

commute pairwise, the Jacobi identity on Lie brackets implies that the Ψ

i

also commute pairwise. The Ψ

i

have coefficients in the algebraically closed field C and commute pairwise, and therefore they are simultaneously triangularizable in a basis pC

j

q of Lie

alg

pA

sub

q . By construction, the C

j

are constant matrices. Each C

j

lies in a characteristic space of Ψ

i

associated with an eigenvalue λ

i,j

. We define λ

_j

p x q : °

_δ

i1

g

_i

p x q λ

_i,j

. As Ψ °

_δ

i1

g

_i

p x q Ψ

_i

, we see that the λ

_j

p x q P k are the eigenvalues of Ψ and that the matrix of Ψ is triangular in the basis p C

_j

q of

Lie

_alg

p A

_sub

; k q .

Remark 3.8. One may refine this proof to predict the eigenvalues of Ψ. Let γ

1

pxq, . . . , γ

ω

pxq P k be the eigenvalues of A

diag

pxq . The above reasoning shows the ex- istence of P

1

P GL

n

C

, such that P

1

A

diag

pxqP

₁¹

:

L

1

p x q 0 . ..

0 L

ω

p x q

, where for

1 ¤ i ¤ ω, L

i

pxq is a matrix in coefficients in k, with only one eigenvalue γ

i

pxq .

In the proof of Lemma 3.7, we have proved the existence of a basis of constant matrices, such that the matrix of the linear map Ψ in this basis is block-diagonal, with blocks that are upper-triangular matrices corresponding to convenient restriction of the linear maps Ψ

_i,j

: X

_i,j

ÞÑ X

_i,j

L

_i

p x q L

_j

p x q X

_i,j

. For 1 ¤ i, j ¤ ω, the map Ψ

_i,j

admits only one eigenvalue that is equal to γ

i

p x q γ

j

p x q P k. Then, the eigenvalues of Ψ are of the form t γ

_i

p x q γ

_j

p x q , 1 ¤ i, j ¤ ω u . Now the diagonal blocks are symmetric powers of A

_1,red

p x q ; the latter has an abelian associated Lie algebra and is triangular. It follows that the γ

i

pxq are linear combinations (with integer coefficients) of the eigenvalues of A

_1,red

p x q , so that the eigenvalues of Ψ also are linear combinations (with integer coefficients) of the eigenvalues of A

_1,red

p x q .

3.4. Decreasing the Dimension of Lie p A; k q . We refer to §2.3, §3.1 and §3.2 for the notations and definitions used in this subsection. The aim of this section is to find a gauge transformation P p x q such that Y

¹

p x q P p x qr A p x qs Y p x q is in reduced form. Thanks to Corollary 3.5, it is sufficient to compute a gauge transformation P p x q P !

Id

n

B p x q , B p x q P Lie

_alg

p A

_sub

; k q )

such that, for every gauge transformation Q r p x q P !

Id

n

B p x q , B p x q P Lie

_alg

p A

_sub

; k q )

, we have Lie p P r A s ; k q „ Lie

Q r r P r A ss ; k .

The k-linear adjoint map Ψ r, A

diag

s : Lie

alg

pA

sub

; kq Ñ Lie

alg

pA

sub

; kq has its eigenvalues λ

₁

p x q , . . . , λ

_κ

p x q in k (see Lemma 3.7) and its minimal polynomial has the form

Π

_Ψ

p X q

¹

κ i1

p X λ

i

p x qq

^mi

, with m

i

P N

.

For each eigenvalue λ

_i

p x q , we let E

_λi

: ker pp Ψ λ

_i

p x q Id

_σ

q

^mi

q be the corresponding characteristic space. So we have the standard decomposition Lie

_alg

p A

_sub

; k q À

_κ

i1

E

_λi

. Of course, the E

_λi

are Ψ-invariant subspaces. Now Lie

_alg

p A

_sub

; k q is also a Ψ-invariant subspace of Lie

alg

p A

sub

; k q . As the E

λi

have each a basis formed of constant matrices (Lemma 3.7), Proposition 3.6 implies that we thus have

Lie

_alg

p A

_sub

; k q à

^κ

i1

E

_λi

£

Lie

_alg

p A

_sub

; k q .

In the reduction process, we may (and will) hence perform a reduction on each E

_λi

separately. So, without loss of generality, we now assume that Ψ has one eigenvalue λ p x q P k and Π

Ψ

p X q p X λ p x qq

^m

, for some m P N

.

As above, we let E

λ

: ker ppΨ λpxqId

σ

q

^m

q and, for i P t0, . . . , mu , let E

_λ^piq

: ker

p Ψ λ p x q Id

_σ

q

ⁱ

. We have the standard flag decomposition E

_λ

À

_m

i1

E

_λ^piq

{ E

_λ^pi1q

. And, last, we recall that for M p x q P E

_λ^piq

{ E

_λ^pi1q

, we have

(3.3) ΨpMpxqq λpxqM pxq M € pxq, with M € pxq P E

_λ^pi1q

.

3.4.1. Reduction on One Level of a Characteristic Space. Let us first pretend that we know a basis C

₁

, . . . , C

_t

of E

_λ^pmq

{ E

_λ^pm1q

(formed of constant matrices C

_i

, this is possible due to lemma 3.7) such that C

s 1

, . . . , C

t

form a basis of g p k q X

E

_λ^pmq

{ E

_λ^pm1q

. This means that C

₁

, . . . , C

_s

could be “removed” by a gauge transformation.

We decompose Apxq as Apxq Apxq ¯

¸

t i1

a

i

pxqC

i

, where Apxq P ¯ E

_λ^pm1q

. Our gauge transformation matrix is of the form P p x q Id

_n

¸

t i1

f

_i

p x q C

_i

with f

_i

p x q P k. As Ψ p C

_i

q λ p x q C

i

C r

i

, with C r

i

P E

_λ^pm1q

, we apply Proposition 3.6 to obtain:

P r A s A ¯ p x q

¸

t i1

f

_i

p x q r C

_i

¸

t i1

a

_i

p x q λ p x q f

_i

p x q f

_i¹

p x q C

_i

.

We see that, in order to achieve reduction in E

_λ^pmq

{ E

_λ^pm1q

, we should have

f

_i¹

pxq λpxqf

i

pxq a

i

pxq for all i P t1, . . . , su.

In other words, the differential equation y

¹

pxq λpxqypxq a

i

pxq should have a rational solution for each i P t 1, . . . , s u .

In practice, we do not know the C

i

nor the a

i

p x q so we now show how to compute them.

Let B

₁

, . . . , B

_t

denote a basis of E

_λ^pmq

{ E

_λ^pm1q

, formed of constant matrices. We will find candidates for the C

i

by computing which combinations of the B

i

may be “removed” from A p x q by a gauge transformation as above. We decompose A p x q as A p x q A ¯ p x q

¸

t i1

b

i

p x q B

i

. There exist (yet unknown) constants c

_i,j

such that B

_i

¸

t j1

c

_i,j

C

_j

, so that:

A p x q A ¯ p x q

¸

t i1

b

_i

p x q

_t

¸

j1

c

_i,j

C

_j

A ¯ p x q

¸

t j1

_t

¸

i1

c

_i,j

b

_i

p x q

C

_j

.

So, the calculation from the previous paragraph shows that there should exist g

_j

p x q P k such that, for j P t 1, . . . , s u , g

_j¹

p x q λ p x q g

_j

p x q

¸

t i1

c

_i,j

b

_i

p x q . The way to find s, the g

_j

p x q and the c

_i,j

is given by Lemma 3.9 (which is proved here for convenience but is well known to specialists).

Lemma 3.9. Let λ p x q , b

₁

p x q , . . . , b

_t

p x q be elements of k. The set of tuples p g p x q , c

₁

, . . . , c

_t

q P k C

^t

such that g

¹

pxq λpxqgpxq

¸

t i1

c

i

b

i

pxq is a C-vector space. Moreover, one can effectively compute a basis of this vector space.

Proof. Let L

_b

be the linear differential operator of order t whose solution space is spanned by b

1

p x q , . . . , b

t

p x q . Let L : L

b

_dx^d

λ p x q

, where the product is the composition, i.e. the usual product in the non-commutative Ore ring k r

_dx^d

s . One readily sees that a function g p x q P k satisfies L p g p x qq 0 if and only if L

_b

p g

¹

p x q λ p x q g p x qq 0, i.e. if there exist constants c

i

P C such that g

¹

p x q λ p x q g p x q

¸

t i1

c

i

b

i

p x q . Hence, the set of tuples pgpxq, c

1

, . . . , c

t

q P k C

^t

such that g

¹

pxq λpxqgpxq

¸

t i1

c

i

b

i

pxq is isomorphic with the set of rational solutions g p x q of L. The latter is a vector space whose basis can be effectively

computed, see §2.3.1.

Lemma 3.9 allows us to compute easily, see §2.3.1, a dimension s P N and a basis p g

_j

p x q , c

_p,jq

q

j1..s

of elements in k C

^t

such that the equation y

¹

p x q λ p x q y p x q

¸

t i1

c

_i,j

b

_i

p x q has a rational solution y p x q g

_j

p x q . The unknown functions a

_i

p x q that we were

looking for are thus given by a

_i

p x q

¸

t i1

c

_i,j

b

_i

p x q .

Via the incomplete basis theorem, we construct a constant invertible matrix Q P GL

t

C whose first s columns are the c

_p,jq

. We may view Q as the base change matrix from the basis pB

j

q

^t_j1

of E

_λ^pmq

{E

_λ^pm1q

to a new basis pC

j

q

^t_j1

of E

_λ^pmq

{E

_λ^pm1q

. Let γ

i,j

denote the entries of Q

¹

.

Lemma 3.10. Let s P N , p g

_j

p x qq

j1,...,s

, and p γ

_i,j

q be computed as in the above paragraph.

For i P t 1, . . . , t u , let f

_i

p x q :

¸

s j1

γ

_i,j

g

_j

p x q . Finally, let P

_λ^pmq

: Id

_n

¸

t i1

f

_i

p x q B

_i

. Then P

_λ^pmq

is a partial reduction matrix, in the sense that

(3.4) Lie

_alg

P

_λ^pmq

r A s ; k X

E

_λ^pmq

{ E

_λ^pm1q

g p k q X

E

_λ^pmq

{ E

_λ^pm1q

. Furthermore, for all Qpxq r : Id

n

¸

t is 1

h

i

pxqC

i

with h

s 1

pxq, . . . , h

t

pxq P k, we have Lie p P

_λ^pmq

r A s ; k q Lie

Q r r P

_λ^pmq

r A ss ; k .

Proof. We apply the first point of Proposition 2.1 (because G is connected, see the proof of Lemma 3.4) to deduce that g p k q „ Lie

_alg

P

_λ^pmq

r A s ; k . Then, (3.5) g p k q X

E

_λ^pmq

{ E

_λ^pm1q

„ Lie

alg

P

_λ^pmq

r A s ; k X

E

_λ^pmq

{ E

_λ^pm1q

.

We want to prove the equality. By construction, C

1

, . . . , C

s

vanish in the construction of P

_λ^pmq

r A s so that C

_{s 1}

, . . . , C

_t

now form a basis of Lie

_alg

P

_λ^pmq

r A s ; k X

E

_λ^pmq

{ E

_λ^pm1q

. Due to Theorem 3.3, there exists R r p x q Id

_n

¸

t is 1

h

_i

p x q C

_i

R p x q , with h

_i

p x q P k, R p x q P E

_λ^pm1q

, such that

(3.6) g p k q X

E

_λ^pmq

{ E

_λ^pm1q

Lie

_alg

R r r P

_λ^pmq

r A ss ; k X

E

_λ^pmq

{ E

_λ^pm1q

. But by construction, we have the inclusion

(3.7) Lie p P

_λ^pmq

r A s ; k q X

E

_λ^pmq

{ E

_λ^pm1q

„ Lie

R r r P

_λ^pmq

r A ss ; k X

E

_λ^pmq

{ E

_λ^pm1q

. Combining (3.5), (3.6) and (3.7) proves (3.4).

Let Q r p x q : Id

_n

¸

t is 1

h

_i

p x q C

_i

with h

_{s 1}

p x q , . . . , h

_t

p x q P k. By construction, we have (3.8) Lie p P

_λ^pmq

r A s ; k q X

E

_λ^pmq

{ E

^p_λ^m1q

Lie

Q r r P

_λ^pmq

r A ss ; k X

E

_λ^pmq

{ E

_λ^pm1q

.