Nonintegrability by discrete quadratures
Guy Casale

Julien Roques

December 19, 2011

Abstract
We give a necessary condition for integrability by discrete quadratures of systems of difference equations : the discrete variational equations along algebraic solutions must have virtuallysolvable Galois groups. This necessary condition à la Morales and Ramis is used in order toprove that q-analogues of Painlevé I and Painlevé III equations are not integrable by discrete
quadratures.
Résumé
Nous nous intéressons à la notion d’intégrabilité par quadratures discrètes pour les systèmes
d’équations aux différences algébriques. Nous montrons que les groupes de Galois des équations
variationnelles discrètes le long des solutions algébriques d’un système aux différences intégrable
sont virtuellement résolubles. Cette condition nécessaire à l’intégrabilité, dans la veine de lathéorie de Morales et Ramis, est ensuite utilisée pour montrer que deux types d’équations dePainlevé discrètes ne sont pas intégrables par des quadratures discrètes.

Contents
1 Frame bundles and prolongations
1.1 RM has a canonical parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 RM is a principal bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 RM is a ‘natural’ bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 C[RM ] is a differential ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4
4
5
5
6

2 Groupoids and Algebroids
2.1 The groupoid EM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 The groupoid AutM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Subgroupoids and pseudogroups . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 The Lie algebroid eM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 The Lie algebroid autM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Some properties of pseudogroups . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6
6
7
7
8
8
9

3 Malgrange’s pseudogroup
3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99114 Galois theory for linear q-difference equations
4.1 Picard-Vessiot theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Tannakian approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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13
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5 A linearization theorem
5.1 Variational equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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MSC 2010 : 39A45, 33D15, 33E17

1

6 Nonintegrability of a discrete Painlevé I equation
6.1 Invariant curve and discrete variational equations . . . . . . . . . . . . . . . . . . .
6.2 Non virtual solvability of the Galois group of the discrete variational equation . . .

16
16
17

7 Nonintegrability of a discrete Painlevé III equation
7.1 Invariant curve and discrete variational equation . . . . . . . . . . . . . . . . . . .
7.2 Non virtual solvability of the Galois group of the discrete variational equation . . .

18
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Introduction
There is no consensus on a general notion of integrability for dynamical systems. G. D. Birkhoff
wrote in [5] “It is a well-known fact that for certain problems, auxiliary analytic relations can be
deduced by means of which the solutions of the system of differential equations can be satisfactorily treated, in which case the system may be said integrable. When, however, one attempts toformulate a precise definition of integrability, many possibility appear, each with a certain intrinsic
interest.”
In this paper, we are interested in the integrability by discrete quadratures of rational systems
of nonlinear σ-difference equations of the form
 σy1 = E1 (t, y1 , y2 , . . . , ym )
..
(1)
.


σym = Em (t, y1 , y2 , . . . , ym )where σ is, for instance, the shift t 7→ t + 1 or the q-dilatation operator t 7→ qt . The system (1) is
integrable by discrete quadratures if it admits a general σ-Liouvillian solution (in the sense of C.
H. Franke in [20]); see Definition 3.11 for details. Roughly speaking, this ensures that there is aformula involving only “elementary operations” to get sufficiently transcendental solutions of (1).
In [9], we studied the integrability in the sense of Mishchenko-Fomenko (Definition 3.3 in loc.
cit.) in the symplectic context. This includes the classical integrability in the sense of Liouville(meaning the existence of sufficiently many linearly independant first integrals in involution; seefor instance section 3 in loc. cit.). There is no inclusion between these notions of integrability andthe integrability by discrete quadratures considered in the present paper.
For more informations about the numerous notions of integrability, we refer the reader to the
expository articles [69, 70, 29, 24].
One of our main results is :Theorem. If (1) is integrable by discrete quadratures then the Galois groups of its discrete variational equations (of any order) along particular algebraic solutions are virtually solvable.

This result is inspired by the so-called Morales-Ramis theory [43, 44, 45] (for its ramifications
and its innumerable applications, see [38, 48] for instance).
The statement of the above theorem involves the Galois groups of linear difference equations(namely, the discrete variational equations) but its proof heavily relies on the Galois theory of
nonlinear difference equations.
Galois’ ideas were first transposed to linear differential equations by E. Picard [51] and E.
Vessiot [71]. Later, they were extended to linear difference equations. A number of authors havecontributed to the development of Galois theories for linear difference equations over the past
years, among whom C. H. Franke [19, 20, 21, 22], P. Etingof [18], M. van der Put and M. Singer[54], M. van der Put and M. Reversat [53], Z. Chatzidakis and E. Hrushovski [12], J. Sauloy [62],
Y. André [2], C. Hardouin and M. Singer [32], etc. The relations between the various Galois
theories for difference equations are partially understood. For this question, we refer the reader
to [13, 30, 31].
The extension of the Galois theory of linear differential equations to nonlinear differential equations was first considered by E. Vessiot in [72] and J. Drach in [16]. In the recent years, the subject
has known a renewal of interest with independent articles by B. Malgrange [40] and H. Umemura[67]. The ideas of these authors have been extended to nonlinear difference equations; we refer to2the work of S. Morikawa [46], S. Morikawa and H. Umemura [47] and A. Granier [25, 26]. For comparison results between the various linear and nonlinear Galois theories for differential equations
we refer to B. Malgrange [40] and H. Umemura [67, 68]. For comparison between the linear andnonlinear Galois theories for difference equations we refer to C. Hardouin and L. Di Vizio [30, 31]
and S. Morikawa and H. Umemura [47]. Almost all these definitions are proved to be compatible.
Comparison between A. Granier and S. Morikawa definitions is still missing and is not used here.
In this paper, we recall and develop A. Granier’s approach based on ideas of B. Malgrange.
Any generalization of differential or difference Galois theory to nonlinear equation uses pseudogroups instead of groups. The definition of algebraic pseudogroups seems to be different in eachapproach : D-groupoids in [40] or Lie-Ritt functors in [67]. As we follow B. Malgrange’s point ofview, our main reference for algebraic pseudogroups will be [40] but most of the results used inthis paper were already known by differential geometers. See the books of V. Guillemin and S.
Sternberg [28] or J.-F. Pommaret [52].
The basic strategy for the proof of the above main Theorem is the following. We prove, using
Artin’s approximation theorem, that the pseudogroup of an algebraic σ-difference equation controls the Galois group of its linearization along any particular solution in the following sense : theLie algebra of the Galois group of the linearized equation is a quotient of a Lie subalgebra of thepseudogroup of the nonlinear equation. Then the Theorem is deduced from the fact that the pseudogroup of a nonlinear σ-difference equation integrable by discrete quadratures is infinitesimallysolvable.
Note that, from a Galoisian point of view, the Liouville integrability implies the commutativityof Galois groupoid and that the integrability by discrete quadratures implies the resolubility of thisgroupoid but the converse implications are false.
This paper also provides applications of our main Theorem to q-Painlevé equations.
Corollary. The q-analogue of Painlevé I equation namely q − A17 from H. Sakai’s classification[61] is not integrable by q-quadratures.
Corollary. The q-analogue of Painlevé III equation namely q − PIII from K. Kajiwara and K.
Kimura [34, 35] is not integrable by q-quadratures.
The discrete Painlevé equations are usually qualified as integrable discrete dynamical systems.
This is not in contradiction with our results: in the previous sentence “integrable” roughly meansthat the discrete Painlevé equations have a “non chaotic behavior” whereas the integrability investigated in this paper means solvability by a controlled formula. By the way many discrete Painlevé
equations have been found by using “integrability detectors” such as singularity confinement, zero
algebraic entropy, diophantine integrability, isomonodromic transformations, etc. Algebraic entropy as defined in [4] (see also [17]) has a special interest in the study of integrability. It is easy toprove that the algebraic entropy of discrete systems integrable by discrete quadratures vanishes i.e.
the sequences of degrees of the iterations has a polynomial growth. In [66], T. Takenawa provedthat discrete Painlevé equations have zero algebraic entropy but questions about integrability bydiscrete quadratures or more generally reducibility of these equations was left open. Recently S.
Nishioka investigated the reducibility of discrete Painlevé equations in [49, 50]. The method proposed in the present paper is more general and systematic than S. Nishioka’s calculations but theconclusion is weaker since integrability by discrete quadratures implies reducibility.
To the best of our knowledge, at the moment, there is no general theory for the discrete Painlevé
equations, except Sakai’s geometric approach in [61] based on previous work by M.-H. Saito et H.
Umemura [60]. For detailed informations about discrete Painlevé equations and integrability, we
refer to the expository articles [29, 24].
This paper is organized as follows. In sections 1, 2 and 3 we recall the definition of Malgrange
pseudogroup and we prove its infinitesimal solvability in the integrable case. In section 4 we recall
useful results regarding the Galois theory of linear difference equations. In section 5 we prove ourmain theorem. In section 6 and in section 7 we prove that q-analogues of Painlevé I and Painlevé
III equations are not integrable by discrete quadratures.

3

1

Frame bundles and prolongations

Let M be a smooth irreducible affine algebraic variety over C of dimension d whose coordinate ringis denoted by C[M ].
Following B. Malgrange [40], some classical constructions from differential geometry will bepresented algebraically by means of their functors of points. We refer the reader to § 2 and § 3 of
V. Guillemin and S. Sternberg’s booklet [28] for the point of view of differential geometry. Their
notations differ from ours (frame bundles are denoted by D(O, M ) and pseudogroups by Γ).
The formal frame bundle of M is the complex proalgebraic variety RM of all formal invertible
maps r : (Cd , 0) → M . More precisely, for any C-algebra A, the A-points of RM , called A-frames
on M , are given by
RM (A) = {locally invertible C-algebra morphisms f : C[M ] → A[[t]]}where A[[t]] := A[[t1 , . . . , td ]] is the ring of formal power series in the d indeterminates t1 , . . . , td (tis a shorthand notation for t1 , . . . , td ). A C-algebra morphism f : C[M ] → A[[t]] is said to be locallyinvertible if its scalar extension fA : A[M ] → A[[t]] induces an isomorphism dfA : J/J 2 → (t)/(t)2where (t) := (t1 , . . . , td ) is the ideal of A[[t]] generated by t1 , . . . , td and where J = fA−1 (t); it is\equivalent to require that fA induces an isomorphism fbA : A[M], J → A[[t]] between the formal
completion of A[M ] = A ⊗ C[M ] along J and the ring of formal power series A[[t]] (see [42,chap.29]).
The algebra C[RM ] of this variety is built up from the differential algebra C{M } generated by C[M ] with d commuting derivations ∂t∂1 , . . . , ∂t∂d by inverting the Jacobian determinant
det ∂t∂ i xj
of a transcendence basis x1 , . . . , xd of C[M ] over C. This ring does not depend
1≤i,j≤don the choice of the transcendence basis.
The projection of RM on M is given on the A-points by f 7→ f /(t).
The ring C[RM ] is filtered by the subrings C[RM ]k of differential polynomials of order lessthan or equal to k. The ring C[RM ]k is the coordinate ring of the order k frame bundle Rk M of
M . For any C-algebra A, the A-points of Rk M are given by
Rk M (A) = {locally invertible C-algebra morphisms f : C[M ] → A[[t]]/(t)k+1 }.
A derivation or an endomorphism of C[RM ] is said to be of degree less than or equal to n if itmaps C[RM ]k in C[RM ]k+n for all k ∈ N.
Example 1.1. If M = Ad then C[M ] = C[x1 , . . . , xd ]. We have
 −1

d
]
C[RM ] = C xα
i | 1 ≤ i ≤ d, α ∈ N [δǫ(j)with ǫ(j) = (0, . . . , 0, 1, 0, . . . , 0) ∈ Nd
where the xα
i ’s are indeterminates and δ = det xi
1≤i,j≤dwhere the 1 is at the jth coordinate. We have  −1 dC[Rk M ] = C xα
i | 1 ≤ i ≤ d, α ∈ N s.t. |α| ≤ k δ
where |α| is the sum of the coordinates of α. The order k frame bundle is


k
ℓ+d
Rk M = Ad × GLd × × A( d ) .
ℓ=2

The projection on M is given by the identification of xi with x0i .

1.1

RM has a canonical parallelism

There is a morphism L from the C-Lie algebra χb of formal vector fields on (Cd , 0) to the C-Lie
algebra of vector fields
b is
Lon RM of degree less than or equal to 1. Indeed, the Lie algebra χb is thethe left C[[t]]-module i C[[t]] ∂t∂ i of (t)-continuous derivations of C[[t]]. This Lie algebra χ
4projective limit lim χn of the finite dimensional C-vector spaces χn = χb ⊗ C[[t]]/(t)n+1 . For any←

C[[t]]

b = lim χn ⊗ A of χC-algebra A, for any A-point Z ∈ χb⊗Ab, the map←

C

RM (A) → RM (A[ǫ]/(ǫ2 ))f 7→ (I + ǫZ) ◦ f,where I is the identity, defines a vector field LZ on RM by means of dual numbers (see [33, p. 80]).
This vector field is called the canonical lift of Z. Note that L is compatible with the Lie bracket :L[Z1 , Z2 ] = [LZ1 , LZ2 ].
One can check that the lifts of non vanishing vector fields at 0 ∈ (Cd , 0) have degree 1
whereas the lifts of vanishing vector fields at 0 ∈ (Cd , 0) have degree 0. The Lie subalgebra(t)bχ = (t1 , ..., td )b
χ of vanishing vector fields at 0 ∈ (Cd , 0) is denoted by χ
b0 . This Lie algebra χ
b0 is
0
0
0
n+1
the projective limit lim χn of the finite dimensional C-vector spaces χn = χb ⊗ C[[t]]/(t).
←C[[t]]

This action of χb gives an isomorphism χb × RM → T RM above RM . The inverse morphism
sends the tangent vector f + ǫg : C[M ] → A[[t]][ǫ]/(ǫ2 ) in Tf RM on (bgA ◦ fbA−1 , f ) in χb × RM . Sucha trivialization of the tangent vector bundle is called a parallelism. At each point f of RM , theimage of χb spans Tf RM and its Lie subalgebra χ
b0 spans the relative tangent Tf (RM/M ).

RM is a principal bundle

1.2

Let Γ be the complex proalgebraic group whose A-points are :Γ(A) = {invertible (t)-continuous A-algebra morphisms g : A[[t]] → A[[t]]}.
The proalgebraic structure is given by the projective limit Γ = limΓk of the complex linear←
algebraic groups Γk whose A-points are given byΓk (A) = {invertible A-algebra morphisms g : A[[t]]/(t)k+1 → A[[t]]/(t)k+1 }.
We have an isomorphism
L(Γ)(A) : Γ(A) × RM (A) → RM (A) × RM (A)M(A)(g, f ) 7→ (f ◦ g, f )so that RM is a Γ-principal bundle over M . The Lie algebra of Γ is χ
b0 . The tangent of L(Γ)
gives a map from the tangent at the identity of Γ to the relative tangent of the projection of
RM × RM → RM onto the first factor along the diagonal :M

Tid L(Γ) : Tid Γ × RM → Tdiag (RM × RM )/RM ∼ T RM.
M

b = limχ0n ⊗ A, one gets LZ.
Evaluating on Z ∈ χ
b0 ⊗A
←

1.3

RM is a ‘natural’ bundle

dp.
Let us consider two A-points p, p′ of M . The formal completion of M along p is denoted by M,
We refer to [33, pp. 194-195] for formal schemes. Let ϕ be an isomorphism from be the formal\\
dp with ring C[M[scheme M,], p to the formal scheme M,
p′ with ring C[M], p′ . We have a naturallift Rϕ of ϕ :dp × RM = RM,\p → RM,\p′ = M,[Rϕ : M,p′ × RM
M

M\
defined, for any f : C[M], p′ → A[[t]], by Rϕ(f ) = f ◦ ϕ. This isomorphism has degree 0. This lift
is called the prolongation of ϕ. Note that R(ϕ1 ◦ ϕ2 ) = Rϕ1 ◦ Rϕ2 .
Let p be a A-point of M and let f be a A-frame at p. The frame f is a morphism f : C[M ] →\A[[t]] such that f /(t) = p thus one can take the formal completion fb : C[M], p → A[[t]]. Let X be5\\a formal vector field at p i.e. I + ǫX is a morphism from C[M], p to (C[M], p)[ǫ]/(ǫ2 ). We have anatural lift RX of X:\p(A) → RM,\p(A[ǫ]/(ǫ2 ))RM,
fb 7→ fbǫ ◦ (I + ǫX)\\
where fbǫ is the algebra morphism from (C[M], p)[ǫ]/(ǫ2 ) to A[ǫ]/(ǫ2 ) defined by fb on (C[M], p)and by fbǫ (ǫ) = ǫ. This lift RX is called the prolongation of X on RM . One has [RX1 , RX2 ] =R[X1 , X2 ]. These prolongations commute with the actions of Γ and χb.

1.4

C[RM] is a differential ring

The fact that C[RM ] is a differential ring is clear from its definition but the differential structure
is not canonical. Let h be a Lie algebra such that χb = h⊕χ
b0 . Then C[RM ] is the h-differential
ring generated by C[M ] where the Jacobian determinant of a transcendence basis of C[M ] over C
is invertible. This ring does not depend on the choice of the transcendence basis.
By duality on gets a differential d : h∗ → ∧2 h∗ . The action of h gives a total derivative
D : C[RM ] → C[RM ] ⊗ h∗ describing the differential structure as it is done in [2, 2.1.2.1.]. It canbe extended into D : C[RM ] ⊗ ∧n h∗ → C[RM ] ⊗ ∧n+1 h∗ by D(f ⊗ h∗ ) = Df ∧ h∗ + f ⊗ dh∗ and,by Jacobi’s identity, one has DD = 0.
Example 1.2. If M is a subvariety of Aq of dimension d defined by an ideal I ⊂ C[x1 , . . . , xq ] andif t1 , . . . , td are coordinates on (Cd , 0) then one can choose h = ⊕i C ∂t∂ i . Let R be the ring


d
C xα
i , |1 ≤ i ≤ q, α ∈ N /DI

where DI is the differential ideal generated by I for the following differential
structure : the|1≤i ≤ q, α ∈ Nd
( ∂t∂1 , . . . , ∂t∂d )-differential structure in the sense of E. R. Kolchin [37] of C xαiα+ǫ(j)

defined by ∂t∂j xα. Using previous notations, we have D : R → R ⊗ h∗ defined, for a ∈ R,
i = xiPd ∂aby Da = i=1 ∂ti dti where dti is the dual bases of h∗ .
Now, we get C[RM ] from R by extension by the inverse of the Jacobian determinant of a transcentendal basis of C[M ] over C. This kind of ring extension has a unique structure of differential
ring extension.
From now on such a h is fixed.

2Groupoids and Algebroids

References for general definitions and elementary properties of groupoids and algebroids are [23]for an algebraic geometry point of view and [39] for a differential geometry point of view. In thissection, we will review specific examples needed in the rest of the paper.

2.1

The groupoid EM

The product EM = RM × RM has a structure of groupoid on RM given by• two projections s and t onto the first and the second factor respectively called source andtarget,• a composition c : EM× EM → EM : the projection on the first and third factors,

t RM s• an identity id : RM → EM : the diagonal,• an inverse inv : EM → EM exchanging the two factors.
These maps have order 0 and satisfy some commutative diagrams [23]. This product situation is
the prototype of groupoid. Subgroupoids of EM are algebraic equivalence relations on RM [39,
Example 1.5].
6

This space is endowed with two commuting prolongation procedures for formal vector fields on
M . The first one, the source prolongation of a formal vector field X on M , denoted by Rs X, isdefined by RX on s∗ C[RM ] and 0 on t∗ C[RM ]. The target prolongation Rt X is defined similarly(mutatis mutandis).
dp to the formal scheme M,[
One can also prolong an isomorphism ϕ from the formal scheme M,p′in two ways. The source prolongation Rs ϕ is defined by Rϕ on s∗ C[RM ] and Id on t∗ C[RM ]. The
target prolongation is defined similarly (mutatis mutandis).
These prolongations are also called right and left translations by analogy with groups in [39,Chap. II, Definition 1.2].

2.2

The groupoid AutM

The group Γ acts diagonally on the product RM × RM = EM by g · (f1 , f2 ) = (f1 ◦ g, f2 ◦ g). The
quotient EM/Γ has two projections on RM/Γ = M inherited from s and t and still denoted by s
and t. The groupoid structure of the direct product RM ×RM = EM induces a groupoid structureon the quotient EM/Γ still denoted by c, id and inv. This quotient groupoid will be denoted by
AutM . Points of this space can be identified with formal maps ϕ between formal neighborhoodsof points in M : let f1 , f2 be two A-frames on M , p and q the A-points defined by f1−1 (t) and\\f2−1 (t) respectively and fb1 A : A[M], q → A[[t]] be the corresponding formal], p → A[[t]], fb2 A : A[Mcompletions, thenRM (A) × RM (A) → AutM (A)(f1 , f2 )7→ (fb1 A )−1 ◦fb2 A

is the quotient map. Because source and target prolongations Rs and Rt defined in section 2.1
commute with the action of Γ, they induce prolongations on AutM still denoted by Rs and Rt .
One can find RM from AutM by choosing a closed point p ∈ M and a formal C-frame r :(Cd , 0) → (M, p) at p. The part Aut(p,M) of AutM above {p}×M for the (source,target) projection
can be identified with RM by means of r. Two such isomorphisms are related by the action of Γ.
The group Γk acts on Rk M in the same way and one can define Autk M to be the quotientof Rk M × Rk M by the diagonal action of Γk . Let us denote by the same symbol πkk+1 theprojection of Γk+1 onto Γk and the projection of Rk+1 M onto Rk M . One has (πkk+1 × πkk+1 )(gk+1 ·(f1,k+1 , f2,k+1 )) = (πkk+1 (gk+1 ) · (πkk+1 (f1,k+1 ), πkk+1 (f2,k+1 ))). The groupoid AutM inherits aproalgebraic structure given by limAutk M .
←This groupoid has a differential structure coming from the differential structure of RM . The
structural group Γ of the principal bundle RM → M is a solvable extension of GLd as it is describedin [52] (see also subsection 5.2). Because of this, it is a special group (see [64, 4.4]) and there exist
rational sections r : M 99K RM of the projection of RM on M , i.e. rational moving frames on M ,which can be used in order to trivialize some bundles :∼d• T M 99K M × h by (p, v) 7→ (p, Tp r(p)

−1

v),

∼

• M × RM 99K AutM by (p, f ) 7→ f −1 ◦ r(p).

The group Γ(C(M )) of gauge transformations acts transitively on rational moving frames and on
the trivializations obtained. Using these identifications one can define the differential structure of
AutM by the connection obtained by the following compositions :D⊗1

C[AutM ] → C[RM ]⊗ C[M ] → C[RM ]⊗ h∗ ⊗ C[M ] → C[RM ]⊗ Ω1(M ) → C[AutM ] ⊗ Ω1 (M )s∗ C[M]

Because of the action of Γ(C(M )), the composition above is independent of r and is well defined.
It gives the total derivation of the differential ring D : C[AutM ] → C[AutM ] ⊗ Ω1 (M ).
s∗ C[M]

2.3

Subgroupoids and pseudogroups

There are two possibilities to define subgroupoids of Aut(M ). The first one is to consider algebraic
subvarieties of Aut(M ) (projective limits of subvarieties of Autk (M )) such that restrictions of7source, target, composition, identity and inverse maps induce a groupoid structure. The second one
is to take limit of groupoids i.e. G = lim Gk with Gk a subvariety of Autk (M ) and a subgroupoid←
for k large enough. These objects are too smooth for our purpose and we have to introduce singularsubgroupoids and pseudogroups following [40, Definition 4.1.1.].
Definition 2.1. A singular (sub)groupoid of AutM is an algebraic subvariety G = lim Gk of←

AutM with Gk a subvariety of Autk (M ) and a subgroupoid of Autk (M − S) for k large enoughand S a closed subvariety of M independent of k. Such a S must be smaller than M . The smallest
S is called the singular locus of G.
From the differential structure one gets the notion of differential subvarieties : a D-variety (forshort) is an algebraic subvariety defined by a differential ideal I (i.e. such that DI ⊂ I ⊗ Ω1 (M )).
Definition 2.2. An algebraic subpseudogroup G of AutM is a singular algebraic groupoid given
by a differential ideal (they are called D-groupoids in [40]) for the differential structure previously
defined.

2.4

The Lie algebroid eM

The Lie algebroid of RM × RM is the relative tangent bundle for the source projection above thediagonal. It is a vector bundle over RM canonically isomorphic to its tangent T RM .
The pull back of the usual Lie bracket of vector fields on the sections of T RM defines a Liebracket on the sections of eM . It is compatible with the graduation i.e. it induces a bracket on thevector fields on Rk M . It is obvious that such objects satisfy the definition of Lie algebroid from[39].
A Lie subalgebroid of eM is a linear subspace of eM in the sense of [27] stable by the bracket.
The relative tangent bundle for the source projection of a singular subgroupoid of EM is a Lie
algebroid.
Example 2.3 ([40]). A foliation on M defined by a linear subspace F ⊂ T M gives by prolongationof sections a linear subspace Rt F |diag ⊂ eM stable under Lie bracket. It is a Lie subalgebroid ofeM .
Example 2.4. [39, 40] An equivalence relation E ⊂ M × M gives by prolongation of sections anequivalence relation on EM i.e. a subgroupoid. Its Lie algebroid is the Lie algebroid defined by
the foliation of M by equivalence classes of E.
Remark 2.5. If a foliation F has no first integrals, it cannot be the foliation by equivalence classesof an equivalence relation. From [23], it is not the Lie algebroid of an algebraic groupoid (evensingular).

2.5

The Lie algebroid autM

The Lie algebroid autM of the pseudogroup AutM is the relative tangent of the source projection
along the identity : id∗ T AutM/s.
Since the identity id : M → AutM is a differential morphism for the differential structures givenby the exterior derivative d on C[M ] and by D on C[AutM ], the tangent bundle along id inherits adifferential structure. Furthermore these structures are compatible with source projection because
D|s∗ C[M] = d, thus the relative tangent bundle along the identity autM has also a differential
structure i.e. it is a C[M ]-module with connection.
On the other hand, the tangent of the quotient map RM ×RM → AutM by the diagonal actionof Γ gives an isomorphism between (T (RM/M )) /Γ and autM . An element of the fiber (autM )pof autM at p is identified with a Γ-invariant vector field on the fiber RMp of RM at p.
Now RMp is a principal homogeneous space for two groups whose actions on RMp commute.
Let A be a C-algebra. The first group is Γ and the action of g ∈ Γ(A) on f ∈ RMp (A) is thefgcomposition g · f : C[M ] −→ A[[t]] −→ A[[t]].

8\The second group is Dif
f(M, p) whose A-points are
\
\
\
Dif
f(M, p)(A) = {invertible (p)-continuous A-algebra morphisms ϕ : A[M], p → A[M], p}\
and the action of ϕ ∈ Dif
f(M, p)(A) on f ∈ RMp (A) is the composition
fbA\\
ϕ · f : C[M ] ֒→ A[M], p −→ A[M], p −→ A[[t]]ϕSince these two actions commute, the Γ-invariant vector fields on RMp are exactly the infinites\\
imal generators of the action of Dif
f(M, p). The Lie algebra of Dif
f(M, p) is the Lie algebraχbM,p of formal vector fields at p. So, an element of (autM )p can be canonically identified with aformal vector field on M at p.
Let G ⊂ AutM be a pseudogroup and let p be a closed point of M not in the singular locus S
of G. The Lie algebroid of G is the relative tangent bundle g = id∗ T G/s of the source projection
along id. Its Lie algebra at p is g(p) = p∗ g ⊂ χbM,p under the identification described above.

2.6

Some properties of pseudogroups

A pseudogroup G is usually an algebraic groupoid only out of a singular locus. Nevertheless it
defines a set theoretical groupoid.
Theorem 2.6. Let ϕ1 , ϕ2 be two composable formal maps in neighborhood of two C-points andlet G be a pseudogroup. If ϕ1 and ϕ2 are in G so is ϕ1 ◦ ϕ2 .
The proof of this theorem can be found in [11, Theorem 1.5]. The main ingredient is Artin’s
approximation theorem (already used in a special case in [45]).
A consequence of this theorem is that for any closed point p of M , G(p,p) = s−1 (p) ∩ t−1 (p) isa group and G(p,M) = s−1 (p) is a G(p,p) -principal bundle over t(G(p,M) ) ⊂ M .
The proof of the previous theorem has also the following consequence proved in [11, Lemma
3.3].
Lemma 2.7. If the Lie algebra of a pseudogroup G at a closed point of M − S is solvable witha sequence of derived algebras of length N then, at any closed point p ∈ M , g(p) is a Lie algebrawith a sequence of derived algebras of length less than or equal to N .

3Malgrange’s pseudogroup

3.1

Definition

A set theoretical pseudogroup on M is a set of analytic invertible maps ϕ : U → V between analytic
open sets of M stable by composition (when defined), inversion, restriction, analytic continuation(under the invertibility condition) [28, Definition 1.2].
Let Φ : M 99K M be a rational dominant map. This map is locally invertible. So, its restrictions
to suitable open subsets are invertible and thus generate a set theoretical pseudogroup denoted by
P GΦ. Such a set of invertible maps describes a subset of AutM by taking all Taylor expansionsof every map at every point where it is defined. This subset of AutM is a set theoretical groupoid
and is still denoted by P GΦ.
Example 3.1. Let n ∈ N and let Φ : C → C be the map defined by Φ(x) = xn . The pseudogroup
generated by Φ is given by the set restrictions of determinations of the following maps on any opena
set where it is invertible : x 7→ ζxn with a ∈ Z and ζ is 1 if a ≥ 0 otherwise ζ is a n−a th root ofunity.
Example 3.2 (Granier [26]). Let Y (qt) = A(t)Y (t) be a linear q-difference equation with A(t) ∈GLn (C(t)). The map Φ : Cn+1 99K Cn+1 defined by Φ(t, Y ) = (qt, A(t)Y ) is dominant. Graphs ofsolutions of this equations are Φ-invariant curves. The pseudogroup
generated by Φis the set of
Qn−1
Q−1
determinations of (t, Y ) 7→ (q n , An (t)Y ) where An (t) is i=0 A(q i t) if n ∈ N and i=n A(q i t) if
n ∈ Z − N.
9Definition 3.3. Let M be a smooth complex algebraic variety and let Φ : M 99K M be a rational
dominant map. The Malgrange pseudogroup M alΦ of Φ is the Zariski closure in AutM of the set
theoretical pseudogroup P GΦ generated by Φ.
We have to prove that such an object is an algebraic pseudogroup.
Proof. – Let Z be the Zariski closure of P GΦ in Aut(M ). Let ϕ ∈ P GΦ be a map definedon an open set U . The pointwise Taylor extension
jϕ :

U
x0

−→ AutM
P ∂αϕ(x−x0 )α7→∂xα (x0 )α!

of ϕ gives an analytic section of the source projection AutM → M .
This section is differential for the differential structure d of O(U ) and D of C[AutM ] i.e. themorphism ϕ# : C[AutM ] → O(U ) is differential. Let I be the ideal of Z. Then E ∈ I can be
written as ϕ# E = 0 for all ϕ ∈ P GΦ. This implies that ϕ# DE = 0 for all ϕ ∈ P GΦ and I isdifferential.
Aut(M ) acts on itself by target composition and one can consider the stabilizer of Z. This is
the largest subvariety G ⊂ Aut(M ) such that there is an hypersurface S ⊂ M and an inclusionc(Z × G) ⊂ Z above (M − S) × (M − S). One can prove the existence of G above the generic pointM

C(M × M ) by elimination. This is proved in [8, pp. 773-774] following classical argument fromalgebraic group theory [65, Proposition 2.2.4]. Then G is a D-subvariety of Aut(M ) above thegeneric point of M × M , by Ritt-Raudenbush basis theorem ([57]) one gets existence of G above
M × M.
Stability of G by composition out of an hypersurface follows from the associativity of thecomposition and thus G ∩ i(G) is a singular subgroupoid of Aut(M ) included in Z. By [40] Se may
be chosen to be some (S × M ) ∪ (M × S) with S a hypersurface of M . Let U and V be two Zariski
dense open subset of M suchthat Φ|U :U → V is a covering. Because Rt Φ|U (P GΦ)|U = (P GΦ)|V ,one has (P GΦ)|V ⊂ Rt Φ|U (P GΦ)|U ⊂ Rt Φ|U (P GΦ)|U ⊂ (P GΦ)|V .
Reading equations of G, we get that, for any p ∈ U , the Taylor expansion jx0 Φ of Φ at x0 as
well as (jx0 Φ)−1 belong to G. So P GΦ ⊂ G ∩ i(G) thus Z ⊂ G ∩ i(G) and Z is a singular groupoid.
Remark 3.4. In the original definition of A. Granier, M al(Φ) was the smallest algebraic pseudogroup containing P GΦ. This definition gives a simplification as the smallest algebraic variety
containing P GΦ is an algebraic pseudogroup.
Example 3.5 ([6, 7]). Let n ∈ N and let Φ : C → C be the map defined by Φ(x) = xn . We geteasily the inclusion
M al(Φ) ⊂ {ϕ | ϕ(x) = αxβ , α ∈ C∗ , β ∈ C∗ , x ∈ s(ϕ)}.
The right hand side is an algebraic pseudogroup in Aut(C) whose ideal is generated as a differential
ideal by xϕϕ′′ − x(ϕ′ )2 + ϕϕ′ .
Example 3.6 (Granier [26]). Let Y (qt) = A(t)Y (t) be a linear q-difference system with A(t) ∈GLn (C(t)) and |q| > 1. Consider Φ : Cn+1 99K Cn+1 defined by Φ(t, Y ) = (qt, A(t)Y ). We haveM al(Φ) ⊂ {(t, Y ) 7→ (αt, B(t)Y ) | α ∈ C∗ , B ∈ GLn (O(U )), U any open set of C}.
This inclusion may be strict. Determination of cases of strict inclusion is the aim of the differential
difference Galois theory developed by C. Hardouin and M. Singer in [32].
Theorem 3.7. Let M and N be two smooth complex algebraic varieties endowed with rational
dominant maps Φ : M 99K M and Ψ : N 99K N . A rational dominant morphism π : N 99K M is adifference morphism if π ◦ Ψ = Φ ◦ π. Assume that such a rational dominant difference morphism
exists. Then, there exists a rational dominant groupoid morphismπ⋆ : M alΨ 99K M alΦ.

10

Proof. – One defines the algebraic pseudogroup Aut(Fπ ) following [40, p 492], as the pseudogroup whose points are the formal invertible π-projectable maps :Aut(Fπ ) = {ϕ ∈ Aut(N ) | ϕ∗ (π ∗ Ω1M ) ⊂ π ∗ Ω1M }= {ϕ ∈ Aut(N ) | ∃π⋆ ϕ ∈ Aut(M ) s.t. π ◦ ϕ = (π⋆ ϕ) ◦ π}.
The projection of such a map induces a mapπ⋆ : Aut(Fπ ) 99K AutM.
By assumption π ◦ Ψ = Φ ◦ π so P GΨ ⊂ Aut(Fπ ) and M alΨ ⊂ Aut(Fπ ). Let π⋆ (M alΨ) be theZariski closure of the image of M alΨ in Aut(Fπ ). Because Ψ is π-projectable on Φ, this is also
true for P GΨ on P GΦ and M alΦ ⊂ π⋆ (M alΨ). But π⋆−1 (M alΦ) is an algebraic pseudogroupcontaining any map π-projectable on a map in M alΦ thus M alΨ ⊂ π⋆−1 (M alΦ). The theorem
follows from this inclusion by applying π⋆ .
Remark 3.8. In usual Galois theory this theorem just states that if Q ⊂ L ⊂ M is a tower of
number fields such that M/Q and L/Q are Galoisian extensions then the small Galois group is aquotient of the big one.

3.2

ExamplesFor basic definition and notation concerning difference algebra, we refer to the book of R.M. Cohn
[14] specially chapter 2 section 4 for notations.
σ-Liouvillian functions
Definition 3.9 (Franke [20, 22]). Let (C(t), σ) be the difference field of rational functions withdifference operator σ a Moebius transformation. A difference extension (K, σ) of (C(t), σ) is said
to be σ-Liouvillian if there exists a tower of differential extensions(C(t), σ) = (K0 , σ0 ) ⊂ (K1 , σ1 ) . . . ⊂ (Kn , σn ) = (K, σ)such that, for all i ∈ {1, ..., n}, the extension Ki−1 ⊂ Ki is either• algebraic,• or additive in the sense that there exist ni ∈ N and zi ∈ Ki such that σini zi − zi ∈ Ki−1 andKi = Ki−1 (zi , . . . , σini −1 zi ),• or multiplicative in the sense that there exist ni ∈ N and zi ∈ Ki such that
Ki = Ki−1 (zi . . . , σini −1 zi ).

nσi i zi∈ Ki−1 andzi

The σ-Liouvillian functions are elements of σ-Liouvillian extensions.
Note that the notion of σ-Liouvillian function is a natural discretization of the usual notion of
Liouvillian function. Indeed, assume for instance that K = C(t) and that σ = σq is the q-dilatation
σq −Idoperator f (t) 7→ f (qt) with q ∈ C∗ . Then, roughly speaking, as q tends to 1, (q−1)ttends to theσ −Idqtends to the usual logarithmic derivative.
usual derivative and (q−1)tP
Assume that K is σ-Liouvillian and that the transcendence degree of K over C is d = 1 + i ni .
Let N be a model for a field K i.e. C(N ) = K. Because this field is a difference field, N is endowedwith a rational map Ψ.

Proposition 3.10. The Lie algebra of the Malgrange pseudogroup of Ψ at p ∈ N is solvable.

11

Proof. – We maintain the notations introduced above. Let m be the smallest common
multiple of ni ’s and z0 = t, z1 , . . . , zd be a transcendence basis in the partial order given by thedefinition. We haveσ m (zi ) = ai (z0 , . . . , zi−1 ) + zi
in the additive case andσ m (zi ) = bi (z0 , . . . , zi−1 )zi
in the multiplicative case for some rational fractions ai and bi with coefficients in K.
By lemma 2.7, it is enough to prove the proposition at a generic point. There exist somespecial rational differential 1-forms θ0 , . . . , θd on W satisfying, for all 0 ≤ i ≤ d, (Ψm )∗ θi = θimod (θ0 , . . . , θi−1 ). The construction of these forms is direct from the definition of the σ-Liouvillian
extensions : we can consider θ0 = dt = dz0 and, for all 1 ≤ i ≤ d, θi = dzi in the additive case oriθi = dzzi in the multiplicative case.
The equations (Ψm )∗ θi = θi mod (θ0 , . . . , θi−1 ) for 0 ≤ i ≤ d are a synthetic way of writing infinitely many algebraic equations satisfied by elements of P G(Ψm ). The Lie algebroid of
M al(Ψm ) at a point p must be included in the solutions of the linearized equations : the vector
fields Y in the Lie algebroid must satisfiy LY θi = 0 mod (θ0 , . . . , θi−1 ) where LY denotes the Liederivative along Y . Let p be a generic point on N and t1 , ..., td be local analytic coordinates suchthat dti = θi mod (θ0 , . . . , θi−1 ). The vector field Y can be written as
Y = c0∂∂∂∂+ c1 (t0 )+ c2 (t0 , t1 )+ · · · + cd (t0 , · · · , td−1 )
.
∂t0
∂t1
∂t2
∂td

The (d + 1)th derived algebra of this type of Lie algebra of formal vector fields is trivial.
The theorem is proved for Ψm . One gets inclusions
M al(Ψm ) ⊂ M al(Ψ) ⊂m−1[

Rs Ψk (M al(Ψm )).

k=0mThis implies that tangent spaces of M al(Ψ ) and M al(Ψ) coincide at smooth points thus the Liealgebroids are equal.

Rational systems integrable by σ-quadratures
Definition 3.11. Let



 σy1
..
.


σym

=

E1 (t, y1 , y2 , . . . , ym )=Em (t, y1 , y2 , . . . , ym )(1)

be a rank m system of nonlinear rational σ-difference equations. This system is said to be integrable by σ-quadratures if it admits a σ-Liouvillian solution (f1 , . . . , fm ) (the coordinates are σLiouvillian) such that C(t, f1 , . . . , fm ) is σ-isomorphic to C(t){y1 , . . . , ym }σ /I where C(t){y1 , ..., ym }σ
is the σ-ring generated by the indeterminates y1 , ..., ym and where I is the σ-ideal generated by
the equations of the system.
In [22] it is proved that if system (1) is linear in y’s then integrability by σ-quadratures is
equivalent to virtual solvability of the difference Galois group. In the nonlinear case one can prove
the analog of one of these implications.
One defines the Malgrange pseudogroup of (1) as that ofΦ:

Cm+1(t, y1 , . . . , ym )

99K
Cm+1
.
7→ (σ(t), E1 , . . . , Em )A consequence of proposition 3.10, theorem 3.7 and lemma 2.7 isTheorem 3.12. If (1) is integrable by σ-quadratures then the Lie algebra of its Malgrange pseudogroup is solvable.
As in the differential, case one can find examples of nonlinear systems with solvable Malgrange
pseudogroups but which are not integrable by σ-quadratures.
12

Galois theory for linear q-difference equations

4

In this section we collect some results concerning the Galois theory of linear q-difference equations
which will be used in the next sections. We set σt = σq t = qt with q ∈ C∗ such that |q| > 1. Tworather different approaches for q-difference Galois theory will be used.

4.1

Picard-Vessiot theory

The construction given here of the Galois group follows the first chapter of [54]. We only assumeπthat σ : P1 → P1 has a Zariski dense orbit. Let G be a complex linear algebraic group. Let E → P1
be a principal G-bundle i.e. the action of G on E gives an isomorphism E × E ∼ E × G over E
P1

for the first projection [64]. For a π-projectable G-equivariant rational dominant map Φ : E 99K E
such that σ = π∗ Φ, P V denotes a closed minimal Φ-invariant subvariety of E dominating P1 andGalΦ its stabilizer in G.
• Two such P V are isomorphic under action of G and called Picard-Vessiot varieties of Φ. The
ring extension C[π(P V )] ⊂ C[P V ] is usually called a Picard-Vessiot extension for Φ.
• The group GalΦ is well defined up to conjugation in G. It is the Galois group of Φ.
• Common level sets of all invariants of Φ in C(E), dominating P1 , are Picard-Vessiot varieties.
Up to some modification of the bundle over ∞ ∈ P1 , we can assume that the bundle is trivial :E = P1 × G. If G = GLn (C), one gets Φ(t, g) = (σt, A(t)g) with A(t) ∈ GLn (C(t)). The equationsof invariants curves g = g(t) are linear σ-difference systems in fundamental form g(σt) = A(t)g(t).
By looking at the first column, one gets the usual vectorial form of linear σ-difference systems :Y (σt) = A(t)Y (t). In this linear situation (E, Φ) will stand either for the system in fundamental
form and (V, Φ) for the system in vectorial form.
From a linear σ-difference system on a vector bundle V on P1 and a point p ∈ C∗ , the funda∗
mental form is given by the action of Φ on the second factor of E = Vp∨ ⊗ V
where Vp is the∨1∗fiber of V at p, Vp is the dual of the trivial bundle Vp × P , ( · ) denotes the open set of fullrank elements and G = GL(Vp ). Then Gal(Φ, p) ⊂ GL(Vp ) is the Galois group defined by the
Picard-Vessiot variety containing Idp ∈ Ep .

4.2

Tannakian approach

In this section, we will only consider q-difference equations (in view of applications given in sections 6 and 7).
Let Dq = C(t)hσ q , σ −1
q i be the non commutative algebra of non commutative polynomials
with coefficients in C(t) satisfying to the relation σ q f = (σq f )σ q for any f ∈ C(t). We denote
by F the neutral Tannakian category over C of q-difference modules over C(t) : it is the fullsubcategory of the category of left Dq -modules whose objects are the left Dq -modules which are
finite dimensional as C(t)-vector spaces. The objects of F can be interpreted as pairs (V, Φ) where
V is a finite dimensional C(t)-vector space V and where Φ is a σq -linear automorphism of V ; fromthis point of view, the morphisms from an object (V, Φ) to an object (V ′ , Φ′ ) are the C(t)-linear
maps F : V → V ′ such that F Φ = Φ′ F ′ .
We define similar categories F (0) and F (∞) by replacing the field C(t) by C({t}) and C({t−1 })respectively. We have natural localization functors F
F (0) and F
F (∞) .
A q-difference module over C(t) is regular singular both at 0 and at ∞ if its localization at 0 has
a lattice over C{t} and its localization at ∞ has a lattice over C{t−1 } which are invariant under
the action of σ q and σ −1
q . We denote by E the full subcategory of F made of its regular singular
objects; it is a neutral Tannakian subcategory of F . For details on what precedes, we refer to M.
van der Put and M. Singer’s book [54] and to J. Sauloy’s paper [62].
For the general theory of Tannakian categories, we refer to [15]. Let ω be a complex valued
fiber functor over F . The Galois group of F is by definition the complex proalgebraic groupπ1q−dif f = Aut⊗ (ω) and the Galois group of an object M of F is the the complex linear algebraic
13

group Gal(M ) = Aut⊗ (ω|hM i) where hM i denotes the Tannakian subcategory of F generated by
M (it is the full subcategory of F whose objects are obtained form M by combining the following∨
operations : tensor products ⊗, direct sums ⊕, duals · , quotients, subobjects).
By Tannakian duality ([15]) ω induces an equivalence of tensor categories between F and therational finite dimensional linear representations of π1q−dif f ; similarly, for any object M of F , ω
induces an equivalence of tensor categories between hM i and the rational finite dimensional linear
representations of Gal(M ). Let M be an object of F and let ρM : π1q−dif f → GL(ω(M )) be therepresentation corresponding to M . We can identify Gal(M ) with the image of ρM (⊂ GL(ω(M ))).
We will freely use the following easy and classical results.
Proposition 4.1. An object M of F is simple if and only if the corresponding rational representation ρM is irreducible if and only if Gal(M ) acts irreducibly on ω(M ).
Proposition 4.2. Let M be an object of F . The determinant of M is trivial if and only if Gal(M )belongs to SL(ω(M )).
We will also use the following result.
Proposition 4.3. If an object M of F has a virtually solvable Galois group then any object of
hM i has a virtually solvable Galois group; in particular, any subobject of M has a virtually solvable
Galois group. We recall that a group if virtually solvable if it has a finite index solvable subgroup.
For algebraic groups it is equivalent to have a solvable Lie algebra.
Proof. – Let N be an object of hM i and let ρN be the rational linear representation of Gal(M )corresponding to N . Since Gal(M )◦ is solvable its image by the rational linear representation ρN ,which is Gal(N )◦ , is also solvable.

Let us now consider a complex valued fiber functor ω (0) over F (0) and take for ω the complex
valued fiber functor obtained by composing ω (0) with the exact, faithful and tensor localization
functor F
F (0) . The local Galois group at 0 of an object M of F is the complex linear algebraic
group Gloc,0 (M ) = Aut⊗ (ω (0) |hM (0) i) (where M (0) denotes the localization of M at 0) which canbe viewed, as above, as a subgroup of GL(ω (0) (M (0) )) = GL(ω(M )).
The localization functor F
F (0) induces, for any object of M of F , a closed immersion (thisis a consequence of [15, Proposition 2.21.] :Gloc,0 (M )֒→ Gal(M ).

The following result Proposition 12.2 in [54]. Compare with Gabber connectedness criterion [36,Proposition 1.2.5].
Theorem 4.4. Let M be an object of F . Then we have a natural surjective morphism :Gloc,0 (M )/Gloc,0 (M )◦ ։ Gal(M )/Gal(M )◦ .
In particular, if Gloc,0 (M ) is connected then Gal(M ) is connected.
The interest of this theorem is that Gloc,0 (M ) is easy to describe if M is an object of E. The
following corollary will allow us to greatly simplify the calculations of Galois groups in sections 6
and 7.
Corollary 4.5. Let M be an object of E. Assume that with respect to some basis the action of σ q
on the localization M (0) of M at 0 is given by a matrix A ∈ GLn (C{t}) such that the eigenvaluesof A(0) belong to q Z then Gal(M ) is connected.
Proof. – Theorem 4.4 ensures that it is sufficient to prove that Gloc,0 (M ) is connected. This
is indeed the case because, in virtue of the description of the Galois groups given in Chapter 12 of[54] of Chapter 2.2 of [62], Gloc,0 (M ) is generated, as a complex algebraic group, by a unipotent
element.
14

In concrete examples, we will work with q-difference systems. Let M be an object of F .
Choosing a C(t)-basis of M , we can interpret M as the q-difference system σq Y = AY where A is
the inverse of the matrix representing the action of σ q on M with respect to the given basis. We
will also work with associated q-difference operators. Concretely, let ΦA be the σq -linear operator
on the n-dimensional C(t)-vector space V = C(t)n given by ΦA (X) = A−1 σq X. We will exhibitn−1e ∈ V such that (e, ΦA (e), ..., ΦA(e)) is a basis over C(t) of V and we will work with P (σ q ) ∈ Dq
where P ∈ C(t)[X] is the unique unitary polynomial of degree n − 1 such that P (ΦA )e = 0; such ae is called a cyclic vector and the theoretical existence of cyclic vector is ensured by the so-called
cyclic vector Lemma ([54, 63, 73]).
We shall conclude this section with the remark that the Tannkian approach gives the same
Galois groups as the usual Picard-Vessiot theory (because the Picard-Vessiot groups can be seenas the tensor automorphisms of some fibre functor over the algebraically closed field C; see [54,§????].

5

A linearization theorem

5.1

Variational equations

Let Φ : M 99K M be a dominant rational map and let C be an algebraic rational Φ-invariant curvewith Φ|C being either t 7→ qt for some q ∈ C∗ with |q| > 1 or t 7→ t + 1. The order k prolongationsof Φ are dominant rational maps Rk Φ on frame bundles Rk M . The restriction of the frame bundles
over C are Γk -principal bundles over C which have a projectable Γk -equivariant map given by the
restriction of Rk Φ over C . This is the order k variational equation in fundamental form. The
projection πkk+1 : Rk+1 M → Rk M obtained from the inclusion of (t)k+1 ⊂ (t)k , with notationsof the first section, is compatible with Rk Φ i.e. Rk Φ ◦ πkk+1 = πkk+1 ◦ Rk1 Φ. Because Rk+1 Φ is
πkk+1 -projectable on Rk Φ, this is also true for Galois groups. We have surjective morphisms
Gal(Rk+1 Φ|C ) ։ Gal(Rk Φ|C ).
We setGal(RΦ|C ) = lim Gal(Rk Φ|C ).
←Theorem 5.1. Let p be a generic point on C . We have a canonical injective proalgebraic groupmorphism
Gal(RΦ|C , p) ⊂ M alΦ(p,p) .
Proof. – By choosing a formal chart at p ∈ C , the Γ-principal bundle RM |C is isomorphic
to the subspace of AutM(p,C ) with source p ∈ C and target in C . Under this identification• AutM(p,p) is Γ,• Gal(RΦ|C , p) is a subgroup acting by left translation.
The closed subvariety M alΦ(p,C ) with source a and target in C of AutM(p,C ) is
• RΦ-invariant because M alΦ and C are RΦ-invariants,• dominates C because its projection contains the orbits of p by Φ which is Zariski dense in C .
This implies that Gal(RΦ|C , p) ⊂ Stab(M alΦ(p,C )) by source composition thus, by Theorem
2.6, Gal(RΦ|C , p) ⊂ M alΦ(p,p) .


5.2

Main theorem

The main theorem is now a consequence of theorem 5.1 and theorem 3.12.
Theorem 5.2. If Φ is integrable by N σ-quadratures then Gal(Rk Φ|C ) is solvable of length N .

15

These groups are algebraic subgroups of Γk (defined in subsection 1.2). They inherit specialkstructure of these groups ; see [52] where they are called GLk (Cd ). Let πk−1
: Γk → Γk−1 be theprojection of order k jets on order k−1 jets and let K be its kernel. The elements of K can be writtenroughly t 7→ t + αtk and one gets that K is a vector group. Then the solvability of a subgroup ofΓk is equivalent to the solvability of its projection in Γk−1 and thus is equivalent to the solvabilityof its projection in Γ1 . So, if the first variational equation has a virtually solvable Galois group
Gal(R1 Φ|C ) then any variational equation has a virtually solvable Galois group Gal(Rk Φ|C ).

6

Nonintegrability of a discrete Painlevé I equation

The rest of this paper is devoted to applications of Theorem 5.2 to concrete discrete Painlevé
equations. See the introduction of the present paper for comments and references about theseequations.
The following system of non linear q-difference equations (qP A′7 in Sakai’s classification [61])is a q-analogue of Painlevé I equation :
1 − xz(x)
y(qx) =z(x)
xy(x)(z(x) − 1)





 z(qx)=

1 − xz(x)xy(x)(z(x) − 1)

2

z(x).

The corresponding dynamical system is :


qx
 


1 − xz
x

z 



Φ : y 7→  xy(z − 1) 
.
2 


1 − xzzzxy(z − 1)

Theorem 6.1. The qP A′7 Painlevé system is non integrable by q-quadratures.
Proof. – It is a consequence of sections 6.1 and 6.2 below and of Theorem 5.2.

Note that a direct proof of nonintegrability of qP A′7 can be found in [49] where the stronger
statement of irreducibility of nonalgebraic solutions is proved.

6.1

Invariant curve and discrete variational equations

Let q4 be a 4th root of q in C. A straightforward calculation shows that Φ leaves globally invariant
the curve parameterized by : 2 t
ϕ : t 7→ q4 /t
1/t
and that Φ acts on the variable t as σq2 , where q2 = q42 , in the sense that :Φ ◦ ϕ = ϕ ◦ σq2 .
This leads to the following discrete variational equation :

q


−1


Y1 (q2 t) = DΦ(ϕ(t))(Y1 (t)) =  q4 t3 (1 − t)−2
q2 t3 (1 − t)
16

0
−1
q2
−2
q43

0





−2t 
q4 (1−t)  Y1 (t)



−3t−1
q2 (1−t)

(2)

6.2

Non virtual solvability of the Galois group of the discrete variational
equationIn this section we shall prove that the q2 -difference system (2) has a non virtually solvable Galois
group. Proposition 4.3 shows that it is sufficient to prove that the following subsystem of (2) has
a non virtually solvable Galois group :"#Y (q2 t) = A(t)Y (t),−2tq4 (1−t)−3t−1
q2 (1−t)

−1
q2
−2
q43

A(t) =∈ GL2 (C(t)).

(3)

This q2 -difference system is actually a basic hypergeometric equation in disguise whose Galois
group was computed in [58].
Let us recall that the q2 -hypergeometric operator with parameters (a, b; c, d) ∈ (C∗ )4 is given
by :



d
c
σq − 1σ q − 1 − t(aσ q2 − 1)(bσ q2 − 1)
q2 2
q2 2




cd
c+d
2
=
− tab σ q2 + −+ t(a + b) σ q2 + (1 − t).
q22
q2
 
0
Coming back to our concrete situation, we claim that e =∈ C2 ⊂ C(t)2 is a cyclic vector for1(3) and that the corresponding q2 -difference operator is q2 -hypergeometric. We have"#
2q43 t3t+1−q2 (1−t)
−1
(1−t)
A =
.
2q4
−q2
Hence
ΦA (e) ="2q43 t
(1−t)

−q2

#

Consequently, e is a cyclic vector and we have"#5
Φ2A (e) =
=

=

A−1

"

2q4 t
(1−q2 t)

−q2

2q3 t

2q5 t

3t+1
4
4
−q2 (1−t)(1−q2 t) + (1−t) (−q2 )2q5 t

2q4 (1−q42 t) + (−q2 )(−q2 )#" 2q3 tq(3t+1)4(1−t) (− (1−q2 t) − q2 )=" 2q3 t=−(.

4q23 t(1−q2 t) + q#

q(3t+1)4(1−t) (− (1−q2 t) − q2 )−q2 (− q(3t+1)(1−q2 t) − q2 )

#

"

0
+
4q3 t
q(3t+1)q2 (− (1−q2 t) − q2 ) + (1−q22 t) + q#

q 3 (1 − t)
2qt + (q + q2 ))ΦA (e) − 2e.
(1 − q2 t)(1 − q2 t)
q3 (1−t)2)
2
So we get the q2 -difference operator σ 2q2 + 2qt+(q+q(1−q2 t) σ q2 + (1−q2 t) . Using the gauge transformation

q3 (1−t)2)22y 7→ ty, we see that σ 2q2 + 2qt+(q+q(1−q2 t) σ q2 + (1−q2 t) is equivalent to σ q2 +2t+(1+q2−1 )σ q2 + (q(1−t). This−1(q2−1 −t)2 −t)is the q2 -hypergeometric operator with parameters (a, b; c, d) = (1, 1; −q2, −1). It is proved in [58]that its Galois group is SL2 (C) (to be precise, in [58] the Galois group computed is that of the operator obtained by permuting (a, b) and (c, d) in the above operator but this is of course inoffensive).

17

7

Nonintegrability of a discrete Painlevé III equation

The following system of non linear q-difference equations is a q-analogue of Painlevé III equation :
1
1 + a0 xz(x)y(qx) =

y(x)z(x) a0 x + z(x)
(4)

−1

1
qa
x
+
y(x)

1
−1

 z(q x) =y(x)z(x) 1 + qa1 x−1 y(x)where a0 , a1 ∈ C∗ . Note that (4) is equivalent to :1
1 + a0 xz(x)y(qx) =

y(x)z(x) a0 x + z(x)




1
a1 x−1 + y(qx)z(qx) =

y(qx)z(x) 1 + a1 x−1 y(qx)
y(x)(a0 x + z(x))(x + a0 x2 z(x) + a0 a1 xy(x)z(x) + a1 y(x)z(x)2 )


.
=

(1 + a0 xz(x))(xy(x)z(x)2 + a0 x2 y(x)z(x) + a0 a1 xz(x) + a1 )The corresponding discrete dynamical system is :



qx
 
1 1 + a0 xz
x




Φ : y  7→ .
yz a0 x + z
 y(a x + z)(x + a x2 z + a a xyz + a yz 2 ) 
z
0
0
0 1
1
(1 + a0 xz)(xyz 2 + a0 x2 yz + a0 a1 xz + a1 )Theorem 7.1. If a0 a1 6∈ −q −N then the qPIII equation (4) is non integrable by q-quadratures.
Proof. – It is a consequence of sections 7.1 and 7.2 below and of Theorem 5.2.

7.1

Invariant curve and discrete variational equation

A straightforward calculation shows that Φ leaves globally invariant the curve parameterized by : t
ϕ : t 7→ 1
1
and that it acts on the variable t as σq in the sense that :Φ ◦ ϕ = ϕ ◦ σq .
The discrete variational equation of Φ along ϕ is easily seen to be given by :


q




0
Y1 (qt) = DΦ(ϕ(t))(Y1 (t)) = 



0

7.2

0

0

−

2
a0 t + 1 Y1 (t)2a0 t + (1 + a0 a1 )t − 3a1 
2a1
t + a1

−1

−



(5)

(a0 t + 1)(t + a1 )Non virtual solvability of the Galois group of the discrete variational
equationIn this section we shall prove that the q-difference system (5) has a non virtually solvable Galois
group. In this purpose, Proposition 4.3 ensures that it is sufficient to prove that the following
18

(regular singular) subsystem of (5) has a non virtually solvable Galois group :
"
#
2a1
−1
t+a1
2
Y (qt) = A(t)Y (t), A(t) =∈ GL2 (C(t)).
2
0 a1 )t−3a1− a0 t (a+(1+a− a0 t+1
0 t+1)(t+a1 )(6)

We claim that the Galois group of (6) is SL2 (C). This assertion will be a consequence of thefollowing three observations.
First observation: G is connected – Indeed, a simple calculation shows that the complex eigenvalues of A(0) belong to q Z . Corollary 4.5 ensures that G is connected.
Second observation: G acts irreducibly on C2 – Indeed, G acts irreducibly if and only if (seeProposition 4.1) (6) is irreducible if and only if some q-difference operator associated to (6) isirreducible over C(t). Let  us now determine an explicit q-difference operator associated to (6).
0
We claim that e =∈ C2 ⊂ C(t)2 is a cyclic vector for (6). Indeed, we have:
1
#
"
2
2a1
0 a1 )t−3a1− a0 t (a+(1+a−−1
t+1)(t+a
)
t+a
0
1
1
,
A =
2
−1
a0 t+1so :−1

ΦA (e) = A



2a1
− t+a
1
σq (e) =−1

is non C(t)-colinear with e. Moreover, we have: 2a1 2
−1 − qt+a1
ΦA (e) = A−1" 2a (a (1+q)t2 +(1+2a a +q)t−2a ) #
1

=

= −

0

0 1

1

(t+a1 )(qt+a1 )(a0 t+1)
a0 qt2 +(a0 a1 +q)t−3a1(qt+a1 )(a0 t+1)
a0 (1 + q)t2 + (1 + 2a0 a1 + q)t − 2a1(qt + a1 )(a0 t + 1)

ΦA (e) −t + a1e.
qt + a1

Hence:L ==σ 2q +

t + a1a0 (1 + q)t2 + (1 + 2a0 a1 + q)t − 2a1
σq +(qt + a1 )(a0 t + 1)
qt + a1σ 2q + α(t)σ q + β(t)is a q-difference operator associated to (6). In order to prove that L is irreducible over C(t), we
follow closely the method presented in [1]. Assume at the contrary that L is reducible over C(t)i.e. that there exist r and s in C(t)× such that :L = (σq − s)(σq − r).
Then we have:r(qt)r(t) + α(t)r(t) + β(t) = 0.
According to [1], r can be decomposed as follows :r=λu(t) c(qt)v(t) c(t)for some λ ∈ C∗ and some unitary polynomials u, v, c ∈ C[t] \ {0} such that:a) ∀n ∈ N, u(t) ∧ v(q n t) = 1;b) u(t) ∧ c(t) = 1;19(7)c) v(t) ∧ c(qt) = 1;d) c(0) 6= 0.
Then equation (7) becomes:
λ2u(t) c(qt)u(qt)u(t) c(q 2 t)
+ λα(t)+ β(t) = 0
v(qt)v(t) c(t)v(t) c(t)(8)

and, clearing the denominators, we get:λ2 (qt + a1 )(a0 t + 1)u(qt)u(t)c(q 2 t)+λ(a0 (1 + q)t2 + (1 + 2a0 a1 + q)t − 2a1 )u(t)v(qt)c(qt)+(a0 t + 1)(t + a1 )v(t)v(qt)c(t) = 0.

(9)

We see that u is a unitary polynomial dividing (a0 t+1)(t+a1 ) and that v is a unitary polynomial
dividing (t + a1 )(a0 q −1 t + 1). Moreover, we claim that we necessary have deg(u) = deg(v). Indeed,
if deg(u) > deg(v) (the case that deg(u) < deg(v) is similar), then we would have the following
inequalities:=deg(λ2 (qt + a1 )(a0 t + 1)u(qt)u(t)c(q 2 t))
2 + 2 deg(u) + deg(c)>=2 + deg(u) + deg(v) + deg(c)max{deg(λ(a0 (1 + q)t2 + (1 + 2a0 a1 + q)t − 2a1 )u(t)v(qt)c(qt)),≥deg((a0 t + 1)(t + a1 )v(t)v(qt)c(t))}
deg(−λ(a0 (1 + q)t + (1 + 2a0 a1 + q)t − 2a1 )u(t)v(qt)c(qt)2−(a0 t + 1)(t + a1 )v(t)v(qt)c(t))−1contradicting (9). Hence, the only possibilities for (u, v) are (1, 1), (t+a0−1 , t+a1 ), (t+a−1
0 , t+qa0 ),−1−1−1(t + a1 , t + a1 ), (t + a1 , t + qa0 ) and ((t + a0 )(t + a1 ), (t + a1 )(t + qa0 )). Properties a) - d) listed−1
above allow us to reduce the possibilities for (u, v) to (1, 1), (t + a−1
0 , t + a1 ) and (t + a1 , t + qa0 ).
We now consider each case separately.

• Case (u, v) = (1, 1). Considering equation (8) at t = ∞, we obtain λ2 q 2deg(c) + λq deg(c) (1 +11−deg(c)or −q −deg(c)−1 . On the other hand, evaluating (8) at t = 0 weq ) + q = 0 i.e. λ = −qobtain λ2 − 2λ + 1 = 0 i.e. λ = 1. So −1 belongs to the q −N : contradiction.
−deg(c)• Case (u, v) = (t + a−1
or −q −deg(c)−1 . On the other
0 , t + a1 ). As above we have λ = −q1
122
hand, evaluating (8) at t = 0 we obtain λ ( a0 a1 ) − 2λ a0 a1 + 1 = 0 i.e. λ = a0 a1 . Hencea0 a1 belongs to −q −Z : contradiction.
−deg(c)• Case (u, v) = (t+a1 , t+qa−1
or −q −deg(c)−1 . On the other0 ). As above we have λ = −q
2 −1
2
−1
hand, evaluating (8) at t = 0 we obtain λ (q a0 a1 ) − 2λq a0 a1 + 1 = 0 i.e. λ = q−11a0 a1 .
Hence a0 a1 belongs to −q −Z : contradiction.

Third observation: G ⊂ SL2 (C). Indeed, this is a direct application of Proposition 4.2 since thedeterminant A is equal to 1.
Hence, G is a connected algebraic subgroup of SL2 (C) acting irreducibly on C2 . The only
possibility is G = SL2 (C).

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Guy Casale
IRMAR UMR 6625 Université de Rennes 1,
Campus de Beaulieu
35042 Rennes Cedex - France
guy.casale@univ-rennes1.fr
Julien Roques
Institut Fourier UMR 5582 Université Grenoble I - CNRS,
100 rue des Maths, BP 74,
38402 St Martin d’Heres Cedex, France
Julien.Roques@ujf-grenoble.fr

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