arXiv:0803.3951v1 [math.DS] 27 Mar 2008

Dynamics of rational symplectic mappings anddifference Galois theory
Guy Casale∗ & Julien Roques†
March 28, 2008

Abstract
In this paper we study the relationship between the integrability of rational symplectic maps and difference Galois theory. We
present a Galoisian condition, of Morales-Ramis type, ensuring thenon-integrability of a rational symplectic map in the non-commutative
sense (Mishchenko-Fomenko). As a particular case, we obtain a complete discrete analogue of Morales-Ramis Theorems for non-integrability in the sense of Liouville.
RésuméDynamique des applications rationnelles symplectiqueset théorie de Galois aux différences
Dans cet article nous étudions les relations entre l’intégrabilité
des applications rationnelles symplectiques et la théorie de Galois deséquations aux différences. Nous donnons une condition galoisienne dutype Morales-Ramis de non-intégrabilité d’une application rationnelle
symplectique au sens non-commutatif de Mishchenko-Fomenko et enparticulier au sens de Liouville .

The first author is supported by ANR project “Intégrabilité en mécaniquehamiltonienne” n◦ JC05 41465. The second one is supported by the ANR project“Phénomène de Stokes, renormalisation, théories de Galois”.
∗
address : IRMAR UMR 6625, Université de Rennes 1, Campus de Beaulieu 35042
Rennes Cedex - France ; E-mail : guy.casale@univ-rennes1.fr†
address :École Normale Supérieure, Département de Mathématiques et Applications
UMR 8553, 45, rue d’Ulm, 75230 Paris Cedex 05 - France ; E-mail : julien.roques@ens.fr

1

Contents
1 Introduction-Organization

3

2 Notations-Hypotheses

4

3 Integrability of a symplectic map

4

4 A non-linear approach
4.1 Malgrange groupoid and its Lie algebra . . . . . . . . . . . . .
4.2 Malgrange groupoid of an integrable symplectic map . . . . .

5585 Linear difference equations
10
5.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.2 Picard-Vessiot theory for linear difference equations . . . . . . 11
6 Linearization of integrable dynamical systems
13
6.1 Adapted curves and discrete variational equations . . . . . . . 13
6.2 Integrability of the discrete variational equation . . . . . . . . 14
7 Integrability and Galois theory
7.1 Statement of the main theorem . . . . . . . . . . . . . . . . .
7.2 First proof of the main theorem : Picard-Vessiot approach . .
7.3 Second proof using Malgrange groupoid . . . . . . . . . . . . .

15
15
16
17

8 Some examples
8.1 Variants when n = 1 . . . . . . . . . . . . . . . . . . . . . . .
8.2 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.5 Example 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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20
20
21
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1

Introduction-Organization

The problem of deciding whether a given continuous dynamical system is
integrable is an old and difficult problem. Many examples arise in classicalmechanics and physics, the three body problem being one of the most famous. In 1999, after anterior works by a number of authors among whom
Kowalevskaya, Poincaré, Painlevé and more recently Ziglin [25], Morales andRamis exhibited in [13, 14, 15] an algebraic obstruction to the integrability ofHamiltonian dynamical systems. This obstruction lies in the differential Galois group of the linearization of the dynamical system under consideration
along a given particular solution (the variational equation). During the lastyears, the method developed by Morales and Ramis have shown its efficiency,
permitting the resolution of numerous classical problems.
Similar questions occur in the study of discrete dynamical systems. For
instance, a rational map f : P2 (C) → P2 (C) being given the typical question
is : does there exist a non-trivial rational invariant of f ? That is, does there
exist H ∈ C(x, y) non-constant such that H ◦ f = H ? In case of a positive
answer, f is called integrable (we refer to section 3 for the general notion of
integrability). Inspired by Morales-Ramis theory, we point out an algebraic
criterion of non-integrability based on difference Galois theory. Our method
involves the linearization of the dynamical system under consideration alonga “particular solution” (that is an invariant parametrized complex algebraic
curve of genus zero ι (P1 (C))) called the discrete variational equation. Froma technical point of view, we present two proofs of our main theorem : thefirst one is based on the difference Galois theory developed by Singer andvan der Put in [23] (there are some complications with respect to the differential case due to the fact that the Picard-Vessiot ring associated to a given
difference equation is not necessarily a domain), whereas the second one isbased on Malgrange theory.
The paper is organized as follows. In sections 2 and 3, definitions andnotations used in the article are given. In section 4 we introduce the Galois groupoid of B. Malgrange and prove that for an integrable system thegroupoid of Malgrange is infinitesimally commutative. But commutativity of
the Malgrange groupoid is not easily checkable. In section 5, we recall somefacts about difference equations from [23]. Section 6 deals with le linearization of the discrete system and its differential invariants along an invariant
curve. The main theorem is proved in section 7 :Theorem 1. Let f be an integrable symplectic map having a φ-adapted curve3ι. Then the neutral component of the Galois group of the discrete variational
equation of f along ι is commutative.
The two proofs are given in this section. The first one deals directly withthe linearized system following the original approach of Morales-Ramis [14].
The second used the linearization to check the commutativity of Malgrange
groupoid. This work is based on a suggestion of Ramis; we are pleased to
thank him.

2

Notations-Hypotheses

Unless explicitly stated otherwise, in the whole article we will use the following notations and conventions :• the (symplectic) manifolds and mappings are complex;• V will denote a complex algebraic variety of even complex dimension
2n (n ∈ N∗ );• C(V ) will denote the field of rational functions over the algebraic varietyV;• ω will denote a non-degenerate closed holomorphic 2-form on V (symplectic form);• f : V 99K V will denote a birational mapping preserving ω that is suchthat f ∗ ω = ω (symplectic mapping of (V, ω));• the symplectic gradient of a rational function H : V 99K C is therational vector field XH satisfying dH = ω(XH , · );• the induced Poisson bracket on C(V ) will be denoted by {H1 , H2 } =ω(XH1 , XH2 ).

3

Integrability of a symplectic map

For the sake of completeness, we first recall the notion of Liouville integrability of a continuous hamiltonian dynamical system.
Definition 1 ([9]). Let H be a rational Hamiltonian on V . It is said tobe integrable in the Liouville sense (or in the Langrangian way) if there is
H1 , . . . , Hn functionally independent first integrals of H (i.e. {H, Hj } = 0)
in involution :4• functionally independent means that their differentials dH1 (z), ..., dHn (z)are linearly independent for at least one point z ∈ V or, equivalently,
for all z in a Zariski dense open subset of V ;• in involution means that, for all i, j in [[1, n]], {Hi , Hj } = 0.
This means that the motion takes place in the fibers of a Lagrangian fibration. This notion of integrability was generalized by Fomenko-Mishchenko[5] (see also [17, 1]) following previous work of Marsden-Weinstein [12].
Definition 2. Let H be a rational Hamiltonian on V . It is said to be integrable in the non-commutative sense (or in the isotropic way) if there is
H1 , . . . , Hn+ℓTindependent first integrals of H i.e. {H, Hj } = 0 such that the
distribution i ker dHi is generated by the symplectic gradients XH1 , . . . , XHn−ℓas vector space over C(V ).
This means that the motion takes place in the fibers of an isotropic fibration. Mishchenko and Fomenko conjectured that this notion of integrability
is equivalent to Liouville’s one. This question will not be discuss here. For aproof in the finite dimensional case, we refer the reader to [21, 2]. A Galoisiancondition for integrability in the isotropic way was obtained by A. Maciejewki
and M. Przybylska in [10]. As an evidence of Mishchenko-Fomenko conjecture, the Galoisian conditions in both cases are the same.
In the context of discrete dynamical systems (iteration of maps) the same
notion of integrability is used.
Definition 3. The symplectic map f is integrable (in the isotropic way) if
it admits n
T+ ℓ first integrals H1 , ..., Hn+ℓ in C(V ), functionally independent
such that i ker dHi is generated by the symplectic gradients XH1 , . . . , XHn−ℓ
The only point in this definition requiring some comments is the notionof first integral. A rational function H ∈ C(V ) is a first integral of f if H isconstant over the trajectories of f , that is H ◦ f = H.

4

A non-linear approach

4.1

Malgrange groupoid and its Lie algebra

Let V be a dimension n smooth complex algebraic variety. Let Rq V denotethe order q frame bundle on V . This bundle is the space of order q jets ofinvertible maps r : (Cn , 0) → V . Such a jet will be denoted by jq (r).
As soon as one gets local coordinates r1 , . . . , rn on an open subset U of
V , one gets local coordinates riα with i ∈ [[1, . . . , n]] and α ∈ Nn such that5

|α| ≤ q on the open set of Rq V of order q frames with target in U. This
bundle is an algebraic variety and is naturally endowed with two algebraic
actions. The first one is the action of the algebraic group Γnq of order q jetsof invertible maps (Cn , 0) → (Cn , 0) by composition at the source. For thisaction Rq V is a Γnq -principal bundle on V .
The second action is the action of the algebraic groupoid Jq∗ (V, V ) of
invertible local holomorphic maps s : (V, a) → (V, b) by composition at thetarget.
Let us recall the definition of an algebraic groupoid on V .
Definition 4. An algebraic groupoid on V is an algebraic variety G with :• two maps s, t : G → V , the source and the target,• a associative composition ◦ : G s ×t G → G,V• an identity id : V → G such that for all g ∈ G
g ◦ id(s(g)) = id(t(g)) ◦ g = g,
• an inverse ◦−1 : V → V such that ∀g ∈ Gs(g ◦−1 ) = t(g) , t(g ◦−1 ) = s(g) , g◦g ◦−1 = id(t(g)) and g ◦−1 ◦g = id(s(g)).
The induced morphisms on the structure (sheaves of ) rings must satisfy thedual diagrams.
The varieties Jq∗ (V, V ) are archetypes of groupoid on V .
Definition 5. Let f : V 99K V be a rational map. The order q prolongationof f is Rq f : Rq V 99K Rq V defined by (Rq f )(jq (r)) = jq (f ◦ r).
The basic objects used to understand the dynamic of f are (rational) first
integrals also called (rational) invariants, i.e. functions H : V 99K C suchthat H ◦ f = H. From a differential-algebraic point of view, invariants of f
are replaced by differential invariants of f .
Definition 6. A differential invariant of order q of f is an invariant of Rq f ,i.e. H : Rq V 99K C such that H ◦ Rq f = H.
Thanks to the canonical projections πqq+1 : Rq+1 V → Rq V , an order qinvariant is also an order q + 1 invariant. Let us denote by Dq Inv(f ) the
set of all the differential invariants of order q of f . This is a C-algebra offinite type. Invariant tensor fields are particular cases of order 1 differential
invariants.
6

Definition 7. Let V be a smooth complex algebraic variety and f : V 99K V
be a rational map. The Dq -closure of f , denoted by Gq (f ), is the subvariety
of Jq∗ (V, V ) defined by the equations H ◦ jq (s) = H for all H ∈ Dq Inv(f ).
Remark. In general, these subvarieties are not subgroupoids but subgroupoids above a Zariski dense open subset of V .
Let J ∗ (V, V ) be the projective limit lim Jq (V, V ) and πq be the projection←of this space on Jq (V, V ).
Definition 8 (Malgrange). Let V be a smooth complex algebraic variety andf : V 99K V be a rational
map. The Malgrange groupoid of f is the subvariety
T −1∗
of J (V, V ) given by πq Gq (Ψ) and it is denoted by G(f ).

Remark. This is not the original definition from Malgrange [11] which isabout foliation. The extension of the definition to discrete dynamical systems
is straightforward. Proposition 2.36 from chapter 3 of [18] (pp 467–469)
allows us to ‘simplify’ the exposition by using directly differential invariants.
Tdb whose coeffida → V,
A solution of πq−1 Gq (Ψ) is a formal map s : V,
cients satisfy all the equations of the Gq (Ψ). It is said to be convergent if theformal map converges. There are well-known solutions of G(f ): the iterates
f ◦n preserve all the differential invariants of f .
For a rational map Φ : P1 (C) → P1 (C), the Malgrange groupoid is computed in [4]. Following this computation the maps with a ‘small’ Malgrange
groupoid are the maps called integrable i.e. maps with a non trivial commutant [19, 24, 3].
Groupoid are not amenable objets. It is much more easy to work with
the infinitesimal part of a groupoid : its (analytic sheaf of) Lie algebra.
For subgroupoids of J ∗ (V, V ) these Lie algebras are algebras of vector fields.
Let us give some notations before the definitions. Let r1 , . . . rn be local
coordinates on V and riα for 1 ≤ i ≤ n and α ∈ Nn with |α| ≤ q be theinduced coordinates. A function on Rq V can be differentiated with respect
to coordinates on Cn by mean of the chain rule i.e. using the vectors fields
X α+ǫ(j) ∂
Dj =
ri
∂riα
i,α
j

↓

where ǫ(j) = (0 . . . 0, 1, 0 . . . 0), one gets a function on Rq+1 V .
PDefinition 9. Let X = i ai ∂r∂ i be a holomorphic vector field on an open
subsetU ⊂ V . The order q prolongation of X is the vector field Rq X =PαDai ∂r∂α on Rq U ⊂ Rq V .
i,α
i

7

Definition 10. Let V be a smooth complex algebraic variety and f : V 99K V
be a rational map. The Lie algebra of the Malgrange groupoid of f is theanalytic sheaf of vector field X on V such that∀q ∈ N and ∀H ∈ Dq Inv(f ), (Rq X)H = 0.
Remark. The original definition from [11] is richer about the structure ofthis sheaf but for the sequel this definition will be sufficient.

4.2

Malgrange groupoid of an integrable symplecticmap

As we already said invariant tensors are examples of order 1 differential
invariants. We will describe these invariants and their lifting up in somecases.
Lemma 1. Let T be a rational tensor field on Rq V . It induces m functions
hiT , 1 ≤ i ≤ m, on Rq+1 V . Furthermore for a vector field X on V one gets
LRq X T = 0 if and only if (Rq+1 X)hiT = 0 ∀1 ≤ i ≤ m.
Let’s see on examples these functions and this property.
Rational function – A rational function h : V 99K C can be lifted to arational function on Rq V . It is clear that if X is a vector field on V and h is
X-invariant then its lifting up is Rq X-invariant.
Rational form – By the very definition, if ω is a p-form on V then it is anppfunction on (T V ) = J1 ((C , 0) → V ). So it induces p functions on R1 V .
One can then lift these functions on higher order frame bundles.
Now if X is a vector field on V and ω is a X-invariant p-form. Let
r1 , . . . , rn be local coordinates on V , one getsω=X

wI dx∧I and X =

n
X
i=1

I∈C(p,n)

ai

∂
∂ri

where C(p, n) stands for the set of length p increasing sequences i1 < . . . < ip
in {1, . . ., n} and dr∧I stands for dri1 ∧ . . . ∧ drip . The lifting up of ω to R1 V
is the np -uple of functions :ǫ(j)hJω (−ri − ri −) =8

X

wI rIJ

J
where J⊂ {1, . . . , n} with #J = p and rI stands for the determinant of thematrix riǫ(j) i∈I .
j∈J

Easy computations giveX
∂ap
drk ∧ dr∧I−{iℓ } ,XwI dr∧I +(−1)ℓ+1 wI∂rk
X ∂X ∂ai ǫ(j) ∂R1 X =ai+r,∂ri∂rk k ∂riǫ(j)X
X ∂ai ǫ(j) ∂r J
I
J
J
rk
.
and (R1 X)(hω ) =XwI rI +
wI
ǫ(j)
∂rk
∂ri
Thus one has hJLX ω = (R1 X)(hJω ).
LX ω =X

For instance, starting from a X-invariant function h on V one gets n + 1
order 1 differential invariants : h itself and h1dh , . . . , hndh .
P ∂Rational vector field – Let v be a rational vector field on V : v =vi ∂riin local coordinates. An order 1 frame determines a basis e1 , . . . , en of Tr V .
The coordinates of v(r) in this basis determine n functions hiv : R1 V 99K C.
Higher order forms – A p-form ω on Rq V determines functions on R1 Rq V .
The induced function on Rq+1 V are the pull-back by the natural inclusion
Rq+1 V ⊂ R1 Rq V .
Theorem 2. The Lie algebra of Malgrange groupoid of an integrable (in theisotropic way) symplectic map is commutative.
Proof. Vector fields of the Lie algebra of Malgrange groupoid must preserven + ℓ independent functions H1 , . . . , Hn+ℓ : XHi = 0. Because they preserve
Hi and the symplectic form ω, they must preserve the symplectic gradients
Xi .
On the subvarieties Hi = ci the integrable map preserves a parallelism
given by XH1 , . . . , XHn−ℓ . This parallelism is commutative : [XHi , XHj ] =XHj Hi = 0 for all 1 ≤ i ≤ n + ℓ and 1 ≤ j ≤ n − ℓ. Because the Lie algebraof vector fields preserving a commutative parallelism is commutative (see for
instance [8]), the theorem is proved.
For Liouville integrable Hamiltonian vector fields, this theorem was already proved along same lines by Ramis [16].

95Linear difference equations

5.1

Basic concepts

For the convenience of the reader, we recall some well-known concepts fromdifference equations theory.
Definition 11. Let φ be a non-periodic Moebius transformation of P1 (C)and E → P1 (C) a rank n vector bundle on the projective line. A rank n
φ-difference system on E is a lift Φ : E → E of φ linear on the fibers. Let U
be a φ-invariant open set of P1 (C) for the transcendent topology. A section
Y : U → E is a solution of Φ if its graph is Φ-invariant.
Using a rational change of variable any Moebius transformation can beconverted either into a translation φ : z 7→ z + h with h ∈ C or into a qdilatation φ : z 7→ qz with q ∈ C∗ , depending on the number of fixed points ofφ on P1 (C) (one in the first case, two in the second case). The corresponding
equations are respectively called finite difference and q-difference equations.
By a gauge transformation, any φ-difference system can be transformed
in a system on a trivial vector bundle. In this case, considering a coordinate
z on the projective line, Φ is described by an invertible matrix with rational
entries :Φ : P1 (C) × Cn → P1 (C) × Cn(z, Y )
7→ (φz, A(z)Y )with A ∈ GLn (C(z)). The equations satisfy by a solution are :Y (φz) = A(z)Y (z).

(1)

In the sequel, vector bundles will be endowed with a fiberwise symplectic
form ω and Φ will preserve this form. By a fiberwise linear change of variables, one can assume that ω is the canonical symplectic J form on fibers.
Such a φ-difference system has a matrix A in Sp2n (C(z)).
What precedes can be rephrased and generalized using the language ofdifference modules; these notions being used in [23], one of our main references for the Galois theory of difference equations with [22], we shall remind
them. A difference ring is a couple (R, φ) where R is a commutative ring
and where φ is a difference operator, that is an automorphism of R. If R = k
is a field, then (k, φ) is called a difference field. A difference module M overa difference field (k, φ) is a k[φ, φ−1]-module, whose k-algebra obtained after
restriction of scalars is finite dimensional. The choice of a particular k-basis
E of M provides a difference system with coefficients in k : φY = AY where10A ∈ Gln (k) is the matrix representing φE in E; the difference system obtainedby choosing another basis E ′ takes the form φY = A′ Y with A′ = (φP )−1 AP
and P ∈ Gln (k). Conversely, starting from a difference system φY = AY ,A ∈ Gln (k) we construct a difference module in an obvious way; the difference module obtained from φY = A′ Y with A′ = (φP )−1AP and P ∈ Gln (k)is isomorphic to those obtained from A.

5.2

Picard-Vessiot theory for linear difference equations

In the whole section we consider a difference field (k, φ) with algebraicallyclosed field of constants (we remind that the field of constants of (k, φ) is
C = {c ∈ k | φc = c}) and :φY = AY(2)
a difference system with coefficients in k (i.e. A ∈ Gln (k)).
We shall remind the construction of the difference Galois group of (2) due
to M. van der Put and M. Singer in [23].
The counterpart for difference equations of the field of decomposition ofan algebraic equation is the Picard-Vessiot extension.
Definition 12. A Picard-Vessiot ring for the difference system (2) is a kalgebra R such that :(i) an automorphism of R, also denoted by φ, which extend φ is given;(ii) R is a simple difference ring;(iii) there exists a fundamental matrix Y for φY = AY having entries in R
such that R = k[(Yi,j )1≤i,j≤n , det(Y)−1 ].
Let us precise the terminology used in the previous definition :• a difference ideal of a difference ring is an ideal I stable by φ. We saythat R is a simple difference ring if it has only trivial difference ideal;• a fundamental matrix Y for φY = AY having entries in R is a matrix
Y ∈ Gln (R) such that φY = AY.
Van der Put and Singer proved in [23] that such an extension exists andthat it is unique up to isomorphism of difference rings. Furthermore, the fieldof constants of the Picard-Vessiot extension is equal to the fields of constantsof the base field.

11

Let us give the general lines of the proof of the existence of the PicardVessiot extension. We denote by X = (Xi,j )1≤i,j≤n a matrix of indeterminates
over k and we extend the difference operator φ to the k-algebra
U = k (Xi,j )1≤i,j≤n , det(X)−1by setting (φXi,j )1≤i,j≤n = A(Xi,j )1≤i,j≤n . Then, for any maximal difference
ideal I (which is not necessarily a maximal ideal) of U, the couple (R =U/I, φ) is a Picard-Vessiot ring for the difference system (2).

Definition 13. The Galois group G of the difference system (2) is the groupof difference automorphisms of its Picard-Vessiot extension over the base field(k, φ). It is a linear algebraic group over the field of constants C.
If A ‘takes its values’ in a particular linear algebraic group, then the
Galois group of (2) is a subgroup of this algebraic group. For later use, let
us reprove this statement in the symplectic case.
Lemma 2. Assume that k = C(z) and that A ∈ Gln (C(z)) is sympletic :t
AJA = Id where J denotes the matrix of the symplectic form. Then there
exists R a Picard-Vessiot extension of (2) containing a symplectic fundamental system of solutions of φY = AY , that is there exists Y a fundamental
system of solutions of φY = AY with entries in R such that t YJY = Id.
The corresponding Galois group belongs to the symplectic group.
Proof. As above, we denote by X = (Xi,j )1≤i,j≤n a matrix of indeterminates
over k and we extend the difference operator φ to the k-algebra
U = k[(Xi,j )1≤i,j≤n , det(X)−1 ]by setting (φXi,j )1≤i,j≤n = A(Xi,j )1≤i,j≤n . Remark that the ideal Ie of U
generated by entries of t XJX − Id
P is a difference ideal. Indeed this ideal ismade of the linear combinations i,j ai,j (t XJX − Id)i,j andPP
φ i,j ai,j (t XJX − Id)i,j =φ(ai,j )(t φ(X)Jφ(X) − Id)i,jPi,j=φ(ai,j )(t (AX)J(AX) − Id)i,jPi,jt=
i,j φ(ai,j )( XJX − Id)i,j .
e so that
Hence, there exists a maximal difference ideal I of R containing I,
U/I is a Picard-Vessiot ring for φY = AY . The lemma follows easily.

We come back to the general non-necessarily symplectic situation. In
contrast with the differential case, the Picard-Vessiot ring is not necessarily
a domain. Let us go further into the study of its structure. Following [23],the Picard-Vessiot ring, say R, can be decomposed as a product of domains: R = R1 ⊕ · · · ⊕ Rs such that :12(i) φRi = Ri+1 ;(ii) each component Ri is a Picard-Vessiot ring for the difference equation :
φs Y = φs−1 A · · · A Y,
(3)
the corresponding Galois group is denoted by G′ .

The proof of the following lemma is left to the reader.
Lemma 3. If the difference system (2) is integrable then it is also the caseof the difference system (3).
Furthermore, following [23], we have a short exact sequence :0 → G′ → G → Z/sZ → 0.

(4)

The following result will be essential in what follows.
Lemma 4. If G′0 is abelian, then G0 is abelian.
Proof. Indeed, the above exact sequence (4) allows us to identify G′ with analgebraic subgroup of G of finite index, so that G0 ⊂ G′0 .

6Linearization of integrable dynamical systems

In this section, starting with a discrete dynamical system f having an invariant curve of genus zero on which f is a Moebius transform, we explain how
one can obtain by a linearization procedure a φ-difference system (finite difference or q-difference system). Differential invariants of f provide in somecases differential invariants for its linearization.

6.1

Adapted curves and discrete variational equations

Definition 14. A curve φ-adapted to f is a rational embedding ι : P1 (C) 99K
V such that f ◦ ι = ι ◦ φ.
Definition 15. Let ι be a curve φ-adapted to f . The difference system
φY = Df (ι)Y over P1 (C) is called the discrete variational equation of falong ι.
By a rational gauge transform, the pull back of the tangent bundle : ι∗ T V
can be assume to be trivial and the variational equation can be written as aφ-difference system on the projective line as stated in the definition.
13

6.2

Integrability of the discrete variational equation

In this section we prove that the discrete variational equation of f along a
φ-adapted curve is integrable in the following sense :Definition 16. A rank 2n difference equation φY = AY , A ∈ SL2n (C(z))
is integrable if it admits n + ℓ independent
rational first integrals h1 , . . . , hn+ℓ
Tsuch that the distribution ker dz ∩ i ker dhi is generated by the fiberwise
symplectic gradients Xh1 , . . . Xhn−ℓ .
There are two points requiring comments :• a function H ∈ C(z, Yi |1 ≤ i ≤ 2n) is a first integral of a difference
system φY = AY if H(φz, AY ) = H(z, Y ).
• The symplectic structure on the fibers is given by the constant canonicalsymplectic structure on C2n .
Our main tool is the generic junior part that we shall now introduce.
Let us consider a rational embedding ι : P1 (C) 99K V . The set of rational
functions on V whose polar locus does not contains ι(P1 (C)) is denoted by
C[V ]ι
Definition 17. The generic valuation νι (H) of a function H ∈ C[V ]ι alongι is defined by :νι (H) = min{k ∈ N | D k H(ι) 6≡ 0 over P1 (C)}.
It extends to H ∈ C(V ) by setting νι (H) = νι (F ) − νι (G), where H = F/Gwith F, G ∈ C[V ]ι .
Definition 18. Let us consider H ∈ C[V ]ι and Tι = ι∗ T V . We defineHι◦ : Tι 99K C, the generic junior part of H along ι, by :Hι◦ = D νι (H) F (ι).
We extend this definition to H ∈ C(V ) by setting H ◦ = F ◦ /G◦ for any
F, G ∈ C[V ]ι such that H = F/G.
The reader will easily verify that the above definitions do not depend on
a particular choice of F, G.
Remark. Junior parts of F at points p ∈ ι(P1 (C)) in the sense of [13]coincide with the generic junior part over a Zariski-dense subset of P1 (C).
The interest of the generic junior part in our context consists in the factthat it converts a first integral of f into a first integral of the variational
equation.
14

Lemma 5. Let us consider H ∈ C(V ) a first integral of f . Then Hι◦ is a
first integral of the variational equation of f along ι.
Proof. Let us consider F, G ∈ C[V ]ι such that H = F/G. The assertion
follows by differentiating vι (F ) + vι (G) times the equality (F ◦ f ) · G =(G ◦ f ) · F .
Therefore, if H1 , ..., Hn+ℓ are n+ℓ first integrals of f , then (H1 )◦ι , . . . , (Hn+ℓ )◦ιare n + ℓ first integrals of the variational equation of f along ι. However, ingeneral, these function are not functionally independent, even if H1 , ..., Hn+ℓare functionally independent. This difficulty can be overcome by using thefollowing lemma due to Ziglin ([25]).
Lemma 6. Let F1 , . . . , Fk be k meromophic functions functionally independent in a neighborhood of ι(P1 (C)). Then there exist k polynomials P1 , . . . , Pk ∈C[z1 , . . . , zk ] such that the generic junior part (G1 )◦ι , . . . , (Gk )◦ι along ι of
G1 = P1 (F1 , . . . , Fk ), . . . , Gk = Pk (F1 , . . . , Fk ) are functionally independent.
Proof. The usual Ziglin Lemma ensures that the assertion is true if one replaces ‘generic junior part’ by ‘junior part at a given point’. The previous
remark allows us to conclude the proof.
Theorem 3. The variational equation of f along a φ-adapted curve ϕ isintegrable.
Proof. Let H1 , . . . , Hn+ℓ be first integrals Tof f with symplectic gradients
XH1 , . . . , XHn−ℓ generating the distribution i ker dHi .
By Ziglin lemma, junior parts of the first n − ℓ first integrals give n − ℓindependent first integrals (H1 )◦ι , . . . , (Hn−ℓ )◦ι for the variational equation.
The independence of the differentials d(H1 )◦ι , . . . , d(Hn−ℓ )◦ι implies the independence of their symplectic gradients X(H1 )◦ι , . . . , X(Hn−ℓ )◦ι . The set offirst integrals is completed applying Ziglin lemma to the 2ℓ remaining first
integrals.
In the next sections we will prove that the integrability of the variational
equation implies severe constraints on its Galois group (more precisely on
the neutral component of the Galois group).

7

Integrability and Galois theory

7.1

Statement of the main theorem

Theorem 4. Let f be an integrable symplectic map having a φ-adapted curve
ι. Then the neutral component of the difference Galois group of the discretevariational equation of f along ι is commutative.
15

In the following sections we give two proofs of our main theorem : thefirst one is based on Picard-Vessiot theory developed by Van der Put andSinger, the second one is based on Malgrange groupoid.

7.2

First proof of the main theorem : Picard-Vessiot
approachLemma 7. Let (k = C(z), φ) be a difference field with field of constants C.
Let H be a first integral of the difference system with coefficients in C(z) :φY = AY . Let R be a Picard-Vessiot extension for this difference system
which is supposed to be a domain. Let Y ∈ Gln (R) be a fundamental systemof solutions with coefficients in R. Then the function :YH : Cn→ C
y 7→ H(z, Yy)

is invariant under the action of the Galois group G.
The action of the Galois group is given by the following : for (σ, y) ∈G×Cn , we set σ ·y = C(σ)y where C(σ) ∈ Gln (C) is such that σY = YC(σ).
Proof. The first point to check is that, for any y ∈ Cn , H(z, Yy) ∈ C. A priori H(z, Yy) belongs to Frac(R), the Picard-Vessiot field, but φH(z, Yy) =H(φz, (φY)y) = H(φz, (A(z)Y)y) = H(φz, A(z)(Yy)) = H(z, Yy) so that
H(z, Yy) is φ-invariant : H(z, Yy) belongs to the field of contant of thePicard-Vessiot field, that is to C. Since H(z, Yy) lies in the field of constantsC, it is invariant by the Galois group : σH(z, Yy) = H(z, Yy). But, since
H has rational coefficients in z and that C(z) is invariant by the action of
the Galois group, we also have, for all σ ∈ G, σH(z, Yy) = H(z, σYy) =H(z, YC(σ)y) = H(z, Y(σ · y)).
Lemma 8. Suppose that the symplectic difference equation φY = AY withcoefficients in C(z) is integrable and that its Picard-Vessiot ring is a domain.
Then the Lie algebra of its Galois group is abelian.
Proof. Let H1 , ..., Hn+ℓ be n + ℓ first integrals insuring integrability of thedifference equation as in definition 16. Lemma 2 ensures that there exists a
Picard-Vessiot extension R containing a symplectic fundamental system ofsolution Y ∈ Sp2n (R). Because
X ∂ YHi
j

∂yj

dyj =

X ∂Hi
j,k

16

∂Yk

Yk,j dyjPi
and invertibility of Y independence of the n + ℓ forms k ∂H
dYk implies∂Yk
independence the n + ℓ differentials dYHi . Moreover from lemma 7 thisfunctions are invariant under the action of the Galois group.
Let X be a linear vector field on C2n in the Lie algebra of the Galois
group.
By the assumption of integrability, all vector field preserving YH1 , ..., YHn+ℓare combinations of the n − ℓ first symplectic gradients:
X=

n−ℓ
X

c i XY H i

i=1

with ci ∈ C(y1 , . . . , y2n ).
Because the Galois group is a subgroup of Sp2n (C), if it preserves a function, it preserves its symplectic gradient too. Thus one gets [XY Hj , X] = 0
for all 1 ≤ j ≤ n + ℓ andn+ℓXcji dYHj
dci =i=1

for some cji ∈ C(y1 , . . . , y2n ).

If Y is a second vector field in the Lie algebraof the Galois group, an easy computation shows that [X, Y ] = 0. This proves
the lemma
Proposition 1. If a linear symplectic difference system is integrable then
the neutral component of its Galois group is abelian.
Proof. Invoking lemma 3, we get that for every integer s the system φY =φs−1 A · · · φA · AY is symplectic and integrable. The lemma follows from thisobservation and from lemma 4 together with lemma 8 (indeed maintaining
the notations of lemma 4, lemma 8 implies that G′0 is commutative because
G′ is the Galois group of a symplectic and integrable system having a domain(Ri ) as a Picard-Vessiot ring).
First proof of theorem 4.
Proposition 1.

7.3

It is a direct consequence of Theorem 3 and

Second proof using Malgrange groupoid

Let f : V 99K V be a rational map and ι : P1 (C) 99K V a φ-adapted curve.
Lemma 9. If the Malgrange groupoid of f is infinitesimally commutative
then the Malgrange groupoid of Rq f is infinitesimally commutative too.

17

Proof. Differential invariants of f give differential invariants for Rq f as described in 4.2. Because Rq f is a prolongation it preserves also the vector
field Dj on Rq V : (Rq f )∗ Dj = Dj . Hence Malgrange groupoid of Rq f must
preserve these vector fields. A direct computation shows that a vector fieldcommutes with all the Dj if and only if it is a prolongation of a vector fieldon V . The compatibility of the prolongation with the Lie bracket proves thelemma.
Let Bq ι be the pull-back by ι of the frames bundle Rq V . The principal
GL2n (C)-bundle B1 ι is the bundle of frames of ι∗ T V on P1 (C). Let Rq ι denote the extension of ι from Bq ι to Rq V . The system given by the restrictionof R1 f to B1 ι is the fondamental discrete variational equation. Its solutions
are the fondamental solutions of the variational equation. The following
lemma follows from the definition of the discrete variational equation.
Lemma 10. The pull-back by R1 ι of differential invariants defined in a neighborhood of ι(P1 (C)) are differential invariants of the fundamental variational
equation.
We will see in the second proof of theorem 4 that some differential invariants are less easy to grab.
Lemma 11. If the difference system φY = AY on a trivial SL2n (C)-principal
bundle on P1 (C) with A ∈ SL2n (C(z)) is integrable then the fibers-tangent
vector fields of the Lie algebra of its Malgrange groupoid commute.
Proof. The proof is follows exactly the proof of theorem 2
Our second proof of theorem 4 rely on the following conjectural result :Conjectural Statement. For linear (q-)difference systems, the action ofMalgrange groupoid on the fibers gives the classical Galois groups.
The similar statement for linear differential equations is true and is provedin [11]. The proof of this conjectural statement in the q-difference case is awork in progress by A. Granier.
Second proof of theorem 4. Lemmas 9 and 10 prove the theorem when thefirst integrals are independent on a neighborhood of the invariant curve.
Let us first assume that the first integrals are defined on ι it i.e. are in
C[V ]ι but not independent on the curve. Let H be a first integral defined ina neighborhood of the invariant curve and k be its generic valuation. The
form dH is invariant and gives n differential invariants of order 1. If k > 1
18

then these invariant vanish above the invariant curve. The differentials ofthese invariants induced order 2 differential invariants. After k iteration ofthis process one gets order k differential invariants but theirs restrictions on
Bk ι give well-defined functions on B1 ι. These functions are induced on B1 ι
from the junior part Hι◦ on ι∗ T V .
By Ziglin lemma one can assume that the junior parts of the first integrals hence the order k invariant induced are independents. One can apply
Lemma 11 to get the infinitesimal commutativity of the vertical part of the
Malgrange groupoid and by the above conjectural statement the almost commutativity of the difference Galois group.
Now let the first integral be any kind of rational functions on V . SuchFa H can be written H = G
with F and G in C[V ]ι . These two functions
induced functions on the frame bundles which are no more invariant. As in
the previous case, the junior parts F ◦ and G◦ of F and G are the restrictionsof functions induced by dF and dG on the frame bundle. Let denote themby hiF dans hiG for 1 ≤ i ≤ n. Because H is invariant, F and G are semi
invariants : F ◦ f = aF and G ◦ f = aG. Let’s have a look at the lift of thesefunctions on R1 V . One has (hjF )(R1 f ) = hjda F + a(hjF ). Let k be the genericvaluation of F along ι then after k lift of the differential as functions on higher
frame bundles one gets functions hαF such that hαF (Rk f ) = . . . + ahαF where
the dots stand for terms vanishing above ι(P1 (C)). The same computation
on G shows that the junior part of H is an invariant of R1 f above ι. The
arguments used to conclude in the first case can be apply in this situation.
This proves the theorem.
8Some examples

In this section we present some examples in the case 2n = 2. In this situation
rational maps f : V 99K V are allowed to be non-symplectic and integrabilityof f means existence of a non-constant rational first integral.

8.1

Variants when n = 1

Following same lines as the proof of the main theorem, one gets.
Theorem 5. Let V be an algebraic variety of dimension 2 and consider arational map f : V 99K V having a φ-adapted curve ι. If f gets a rational first
integral then the intersection of the neutral component of the Galois group of
the discrete variational equation of f along ι with SL2 (C) is commutative.
19

Proof. The proof follows the same lines as that of the main theorem (mores
simple actually).
We also have the following useful result.
Theorem 6. Let V be an algebraic variety of dimension 2 and considera rational functionf : V 99K V having a φ-adapted curve ι. If one of thefollowing conditions relative to the Galois group G of the discrete variational
equation of f along ι holds then f is not integrable :- diag(C∗ , C∗ ) is conjugated to a subgroup of G;
1 C
is conjugated to a subgroup of G.
0 C∗
Proof. Use the same arguments as for the proof of the main theorem and thefact that a rational function F (z)(x, y) invariant by the natural action of theabove groups (on (x, y)) is necessarily constant.

8.2

Example 1

Considerf:

C2 99K
 
x
7→
y

C2
qx 1− 1 y

x−1

!

y
qy(1 − x−1))

where q, q ∈ C∗ . The curve ϕ(x) = (x, 0) is a σq -adapted to f . The variational equation of f along ϕ is given by :x
q q x−1
Y (qx) =Y (x).
(5)
0
q


1 C
so that f is not
If q is in general position then its Galois group is
0 C∗
integrable. A contrario, remark that when qq is a root of the unity thenthe map f is integrable (F (x, y) = (xy)n is a first integral for n ∈ N large
enough).

8.3

Example 2

Considerf : C2 99K C2qx + ya(x)
x
7→
yb(x)
y
20

with b 6= 0. The curve ϕ(x) = (x, 0) is a σq -adapted to f . The variational
equation of f along ϕ is given by :
q a(x)
Y (x).
(6)
Y (qx) =0 b(x)
x−1
Let us for instance study the case where a(x) = b(x) = q bx−c/qwith b/c 6∈ q Z .
Remark that
!
!
−1



x−1
q q bx−c/q
0
1
0 1
0 1
−1
,
z
= qz
(1+b)z−(1+c/q)
z−1
x−1
−1 1
−1 1
− bz−c/q
0 q bx−c/q
bz−c/qso that (6) is rationally equivalent to (the system associated to) a basic
hypergeometricequation with parameters (1, b, c), the Galois group of which
1 C(see [20]). Consequently, f in not integrable.
is
0 C∗

8.4

Example 3

Considerf : C2 99K C2
x
ya(x) − x + 17→yxyb(y)

where a ∈ C(x) and b ∈ C(y) without pole in 0. The curve ϕ(x) = (x, 0) is a
τ−1 -adapted curve with τ−1 (z) = z − 1. The variational equation of f along
ϕ is given by :
−1 a(x)Y (1 − x) =Y (x).
0 xb(0)

The algorithm exposed in [7] allows us to exhibit many non integrable cases.

8.5

Example 4

Considerf : C2 99K C2qx + q̃y + (x − y)a(x, y)x7→
qx + q̃y + (x − y)a(x, y))ywhere a, a ∈ C(x) and q, q̃, q, q̃ ∈ C are such that q = q + q̃ = q + q̃ ∈ C∗ .
The curve ϕ(x) = (x, x) is a σq-adapted to f . The variational equation of falong ϕ is given by :
q + a(x, x) q̃ − a(x, x)
Y (x).
(7)
Y (qx) =q + a(x, x) q̃ − a(x, x)
21

This system is rationally equivalent to :q − q + a(x, x) − a(x, x) 0
Y (qx) =Y (x).
q
q + a(x, x)(8)One can see that generally f is not integrable.

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