Galois groupoid and confluence of difference equations
aa r X i v : . [ m a t h . AG ] J un GALOIS GROUPOID AND CONFLUENCE OF DIFFERENCE EQUATIONS
GUY CASALE AND DAMIEN DAVY
Abstract.
In this article we compute Galois groupoid of discret Painlev´e equations. Our maintool is a semi-continuity theorem for the Galois groupoid in a confluence situation of a diffrenceequation to a differential equation. Introduction
The aim of this article is to determine Galois groupoids of some discrete dynamical systemsΦ : M M . The phase space M is an algebraic varaiety and Φ is a rational dominant map.Galois groupoid is the “differential algebraic” envelope of Φ, it describes the richest algebraicgeometric structure (see [Gro88]) invariant by Φ.For linear differential equation, E. Picard [Pic87], E. Vessiot [Ves04] and later E.R. Kolchin[Kol73] developed a Galois theory. To an order n linear differential equation with coefficient in thedifferential field (cid:0) C ( x ) , ∂∂x (cid:1) is associated an algebraic sub-group of GL n ( C ). This group measuresalgebraic relations between a basis of solutions and theirs derivatives. This theory interacted withmany different areas of mathematics see [Ber92] or [vdPS03]. A discrete analog was proposed byFranke [Fra63] for linear difference equations (see also [vdPS97]).In [Mal01], a similar object was defined by B. Magrange for foliations, including nonlineardifferential equations. Malgrange called it the Galois groupoid of the foliation. In the presentarticle, we present an extension of this definition to different dynamical systems. Galois groupoidof a rational dominant map is defined and first examples are studied in [Cas06] and [Gra11]. Letus give a quick, unformal and non complete definition, (see section 2 for the formalisation).Let B be a smooth irreducible algebraic variety with en rational dominant self map σ : B B .It corresponds to a difference field of rational functions ( C ( B ) , σ ∗ ). Let M be a smooth irrreduciblealgebraic varierty with a projection π : M → B and Φ : M M a rational dominant lift of σ .The difference equation associated to these datas is the equation y ◦ σ = Φ ◦ y on local holomorphicsections of π , y : U → M , U ⊂ B . Galois groupoid of Φ (or its difference equation) is denoted by M al (Φ /B ), roughly speaking it is the algebraic pseudogroup of local holomorphic transformationsof fibers of π generated by restrictions of Φ to fibers . By considering all order k Taylor expansionsof all elements in
M al (Φ /B ) at all theirs points of definition one gets a finite dimensional algebraicvariety called M al k (Φ /B ). The Galois groupoid has a natural structure of pro-algebraic variety.1.1. Confluence and specialisation.
When a dynamical system depends on parameters, wewonder how the Galois groupoid depends on these parameters. For linear differential equation ananswer was given by [Gol57]. A generalisation of these specialisation results including differenceequation can be found by a Tannakian approach in [And01]. As a byproduct, these theorems canbe used to obtained bound on the Galois group of a differential equation given as the continuouslimite of a difference equation.Let B S be a rational map invariant by σ . We can choose a value s ∈ S and considere therestriction of the dynamical system Φ s : M s M s above σ s : B s B s . The general opinionis that the dynamical behaviour of Φ s for a particular value of s ∈ S must be simpler that thebehaviour on the generic fiber. This is the specialisation theorem. Theorem (7 page 6) . In the situation above, dim
M al k (Φ s /B s ) ≤ dim S M al k (Φ /B ) with equalityfor general s ∈ S . Mathematics Subject Classification.
Key words and phrases. (eng) difference equations, continuous limit, Galois groupoid. (fr) ´equations auxdiff´erences, confluence, groupo¨ıde de Galois.
A property is true for a general s ∈ S if it is true for s out of a countable union of properalgebraic subvarieties of S .The second theorem concerns vector fields obtained as continuous limits of discrete dynamicalsystems. This means that S is one dimensional and near a special point s ∈ S , one can writeΦ s = Id + ( s − s ) X + o ( s − s ) ; Id is the identity map on M s and X is a rational vector fieldon M s . Again we will prove that the dynamical behaviour of X is simpler that the behaviour ofΦ. This is the confluence theorem. Theorem (8 page 6) . In the situation above, dim
M al k ( X/B s ) ≤ dim S M al k (Φ /B ) . The main example.
A discrete Painlev´e equation is a birational self-map Φ of C × C fiberedabove an automorphism without periodic points of the curve C of the independent variable n . Thiscurve may be non compact. Formally the difference equation of invariant curves paramaterized by n is the difference equation. Following [RGH91], two properties are imposed to be called “discrete-Painlev´e equation” :(1) It must have the singularities confinement property. There exists a fiberwise extension M of C × C → C such that Φ can be extended as a biholomorphism of M .(2) It must degenerate on a differential Painlev´e equation. There exists a deformation f M of M above ( C ,
0) and a deformation e Φ of Φ such that for ǫ ∈ ( C ∗ ,
0) the couple ( f M ǫ , e Φ ǫ )is birrational to ( M, Φ) and at 0 one has a Taylor expansion e Φ ǫ = Id + ǫX + o ( ǫ ) with X a vector field on f M whose trajectories parameterized by open set in C are Painlev´etranscendents. This vector field is called a continuous limit of Φ.In [Sak01], H. Sakai achieved the classification of total spaces M of definition of discrete Painlev´eequations. For a overview on difference Painlev´e equations the reader may see [KNY17]. Let usdescribe the easiest non trivial example.1.2.1. The discrete Painlev´e 2 equation.
Considere the rational dominant map :Φ ii : C C nxyabc n + 1 − y + ( a + bn ) x + c − x xabc . Invariant analytic curves parameterized by n are solutions of the so-called discrete Painlev´e twoequation : dP ii ( a, b, c ) : x ( n + 1) + x ( n −
1) = ( a + bn ) x ( n ) + c − x ( n ) . This equation degenerates on the second Painlev´e equation. For ǫ ∈ C ∗ , the change of variable t = nǫ , f = y/ǫ , g = ( x − y ) /ǫ , α = ( a − /ǫ , β = ( b − ǫ ) /ǫ and γ = c/ǫ can be used to get atrivial family Φ II ( ǫ ) of rational dominant maps above C ∗ with a special degenerated fiber at 0. Adirect computation gives the expression of Φ ii in these new coordinatesΦ ii ( ǫ ) : C C tgfαβγ tgfαβγ + ǫ f + tf + γg + o ( ǫ ) . from which one gets the confluence of the family Φ ii ( α, β, γ, ǫ ) on the vector field X γ = ∂∂t + g ∂∂f +(2 f + tf + γ ) ∂∂g whose trajectories parameterized by t are solutions of the second Painlev´e equation P ii ( γ ) : d fdt = tf + 2 f + γ . ALOIS GROUPOID AND CONFLUENCE OF DIFFERENCE EQUATIONS 3
The second Painlev´e equations.
From [CW18], we know the Galois groupoid of X and from[Dav] we know that that Galois groupoid of X γ over C is the pseudogroup of invariance of the timesform dt and the closed 2-form ι X γ dt ∧ df ∧ dg for very general values of γ . The main theorems, Theorem 7 and
Theorem 8 , can be used to determine for very general values of ( a, b, c ) theGalois groupoid of Φ ii ( a, b, c ) over C ( n ): Theorem (9 page 7) . M al (Φ ii ( a, b, c ) / C ( n )) = { ϕ : ( C , p ) → ( C , p ) | p , p ∈ C , ϕ ∗ dx ∧ dy = dx ∧ dy } Galois groupoid and irreducibility.
The notion of irreducibility of a differential equationis as old as differential equations. It is formalized by K. Nishioka [Nis88] and Umemura [Ume88,Ume90] following original ideas from Painlev´e’s Stochkolm Lessons [Pai73]. A discrete version ofirreducibility can be found in [Nis10]. Let us recall a weaker version of the definition.
Definition 1.
A rational second order difference equation E : x ( n + 2) = F ( n, x ( n ) , x ( n + 1)) isreducible if there exist a tower of field extension C ( n ) = K ⊂ K . . . ⊂ K N such that (1) each intermediate extension K i − ⊂ K i is of one of the following type (a) algebraic, (b) generated by entries of a fundamental solution of a linear system with coeffiecients in K i − , (c) generated by a solution of a first order non linear difference equation with coefficientsin K i − , (2) there exists a solution x ∈ K N with n, x ( n ) , x ( n + 1) algebraically independent in K N . From numerous papers by S. Nishioka [NN13, Nis09, Nis12], we know the irreducibility of manydiscrete Painlev´e equation. The proofs are done by direct and very specific computation.The aim of our work is to provide a new proof of irreducibility of most of the discrete Painlev´eequations based on the computation of theirs Galois groupoids using the confluence of such discreteequations on differential Painlev´e equations. Such a proof should prove ireducibility of any discretedynamical system which admit as contiuous limite second order equation with a big enough Galoisgroupoid (such as 1.2.2). This seems to be done using theorems of this article together with discreteanalog of results from [Cas09]. This will be done in future work.2.
Fibered dynamical systems and transversal Galois groupoid
When a dynamical system Φ : M M comes from a difference equation, it preserves thefoliation by fibers of the map M → B given by the independant coordinates. Moreover in the casewith parameters, Φ preserves the fibers of the map given by the parameters. Let M be the phasespace, B the space of the independant variable and parameters and S the parameters space. Wehave a commutative diagram M Φ M ↓ ↓ B σ B ↓ ↓ S = S where σ is the map corresponding to the operator involved in the difference equation.2.1. The fibered frames bundle.
Let q be the dimension of fibers of M over B . The space offrames on fibers of M → B is R ( M/B ) = { r : ( C q , → M b | b ∈ B and det( Jr ) = 0 } where Jr is the Jacobian matrix of r at 0. Its coordinates ring of R ( M/B ) is(
Sym ( C [ M ] ⊗ C [ ∂ , . . . , ∂ q ]) / L ) [1 /jac ]where • the tensor product is a tensor product of C -vector spaces; ALOIS GROUPOID AND CONFLUENCE OF DIFFERENCE EQUATIONS 4 • Sym ( V ) is the C -algebra generated by the vector space V ; • Sym ( C [ M ] ⊗ C [ ∂ , . . . , ∂ q ]) has a structure of C [ ∂ , . . . , ∂ q ]-differential algebra via the rightcomposition of differential operators; • the Leibniz ideal L is the C [ ∂ , . . . , ∂ q ]-ideal generated by f g ⊗ − ( f ⊗ g ⊗
1) for all( f, g ) ∈ C [ M ] , h ⊗ ∂ i for h ∈ C [ B ] and 1 ≤ i ≤ q and 1 ⊗ − • the quotient is then localized by jac the sheaf of ideals (not differential !) generated bydet ([ x i ⊗ ∂ j ] i,j ) for a transcendental basis ( x , . . . , x q ) of C ( M ) over C ( B ) on Zariski opensubset of M where such a basis is defined.Local coordinates ( b , . . . b p , x , . . . , x q ) on M such that the projection on B is the projectionon b ’s coordinates, induce local coordinates on R ( M/B ) via the Taylor expansion of maps r at 0 : r ( ǫ . . . , ǫ q ) = (cid:18) b , . . . b p , X r α ǫ α α ! , . . . , X r αq ǫ α α ! (cid:19) . One denotes x αi : R ( M/B ) → C the function defined by x αi ( r ) = r αi . This function is the element x i ⊗ ∂ α in C [ R ( M/B )].(1) The action of ∂ j : C [ R ( M/B )] → C [ R ( M/B )] can be written in local coordinates and givesthe total derivative operator P i,α x α +1 j i ∂∂x αi where 1 j is the multiindex whose only nonzero entry is the j th and its values is 1.(2) The vector space C [ ∂ , . . . , ∂ q ] is filtered by C [ ∂ , . . . , ∂ q ] ≤ k the spaces of operators of orderless than k . This gives a filtration of C [ R ( M/B )] by C -algebras of finite type.(3) These algebras are coordinate ring of the space of k -jet of frames R k ( M/B ) = { j k r | r ∈ R ( M/B ) } .(4) The action of ∂ , . . . , ∂ q have degree +1 with respect to the filtration.2.2. Prolongation of dominant morphism and vector fields.
Morphisms (resp. derivations)from C [ M ] to C [ M ] with a non zero Jacobian determinant and preserving C [ B ] act on C [ R ( M/B )]as morphisms (resp. derivations).If the map Φ is regular and induces Φ ∗ : C [ M ] → C [ M ]. Its action of the frame bundle isdefined by Sym (Φ ⊗
1) on
Sym ( C [ M ] ⊗ C [ ∂ , . . . , ∂ q ]), it can be easily checked that the Leibnizideal is preserved. The induced map C [ R ( M/B )] → C [ R ( M/B )] corresponds to a endomorphismof R ( M/B ) denoted by R Φ and is called the prolongation of the morphism. The prolongation ofa derivation of C [ M ] preserving C [ B ] is a derivation of C [ R ( M/B )] defined in the same way.If the map Φ is rational and dominant with domain of definition M ◦ , its prolongation on R ( M ◦ /B ) is a rational dominant map on R ( M/B ). Rational vector fields on M can be pro-longed on R ( M/B ) in a similar way.Prolongations of morphisms and derivations of C [ M ] on C [ R ( M/B )] have degree 0 with respectto the filtration defined above. Prolongations commute with the differential structure. When X isa rational vector field on M preserving M → B , its prolongation RX on R ( M/B ) can be computedwith a explicit formula :Let X = P c i ( b ) ∂∂b i + P a i ( x, b ) ∂∂x i be a vector field preserving M → B in local coordinates.One gets RX = P c i ( b ) ∂∂b i + P i,α ∂ α ( a i ) ∂∂x αi . The space R ( M/B ) is a principal bundle over M . Let us describe this structure here.The pro-algebraic group Γ = { γ : \ ( C q , ∼ → \ ( C q , } is the projective limit of groups Γ k = { j k γ | γ ∈ Γ } . It acts on R ( M/B ) by composition and R ( M/B ) × Γ → R ( M/B ) × M R ( M/B ) sending ( r, γ ) to ( r, r ◦ γ ) is an isomorphism. The actionof γ ∈ Γ on R ( M/B ) is denoted by Sγ : R ( M/B ) → R ( M/B ) as it acts as a change of sourcecoordinate of frames. At the coordinate ring level, this action is given by the action of formalchange of coordinate on C [ ∂ , . . . , ∂ q ] followed by the evaluation at 0 in order to get operators withconstant coefficients. This action has degree 0 with respect to the filtration induced by the orderof differential operators meaning that for any k , the bundle of order k frames R k M is a principalbundle over M for the group Γ k = { j k γ | γ ∈ Γ } . ALOIS GROUPOID AND CONFLUENCE OF DIFFERENCE EQUATIONS 5
Compatibility of the differential and principal structures.
Let ǫ , . . . ǫ q be coordinateson ( C q ,
0) such that ∂ i = ∂∂ǫ i . The Lie algebra of Γ is the Lie algebra of formal vector fields vanishingat 0 : b X ( C q , v = P a i ( ǫ ) ∂ i ∈ b X ( C q , R ( M/B ) is f ⊗ P f ⊗ ( P ◦ v ) | ǫ =0 . When v belongs to L C ∂ i , this action is thedifferential structure ; when v belongs to b X ( C q , The fibers automorphism groupoid.
A groupoid is naturally associated to any principalbundle. In the case of R ( M/B ) one can give three definitions of this groupoid.(1) The groupoid
Aut ( M/B ) is the groupoid of formal invertible maps: { b ϕ : ( M b , p ) ∼ → ( M b ′ , p ) | ( b, b ′ ) ∈ B × B, p ∈ M b , p ∈ M b ′ } (2) The map R ( M/B ) × R ( M/B ) → Aut ( M/B )( r, s ) r ◦ s ◦− realized the quotient by the diagonal action of Γ.(3) The groupoid of Γ-equivariant maps between fibers of π : R ( M/B ) → M is Aut ( M/B ).The second definition is adapted to prove that
Aut ( M/B ) is pro-algebraic groupoid.2.6.
Galois groupoids.
It is a proper generalisation of the Galois group of linear differential ordifference equation. It is a quotient of the Galois groupoid defined in [Mal01] for foliations and in[Gra11, Cas06] for rational dominant maps.2.6.1.
Galois groupoid of a rational vector field.
The Galois groupoid for vector fields where alreadystudied in [Dav]. We recall the definition in the fibered situation. The proof are immediate from[Dav] or [CD20] and we left them to the reader.A rational functions H ∈ C ( R ( M/B )) such that RX · H = 0 is a differential invariant of X . Let Inv ( X ) ⊂ C ( R ( M/B )) be the subfield of differential invariants of X . Let W be a model for Inv ( X )and π : R ( M/B ) W be the dominant map induced by the inclusion Inv ( X ) ⊂ C ( R ( M/B )).
Definition 2.
The transversal Galois groupoid is
M al ( X/B ) = R ( M/B ) × W R ( M/B ) ⊂ Aut ( M/B )To defined properly this fiber product one needs to restrict π : ( R ( M/S )) o → W on its domain ofdefinition then ( R ( M/B ) × W R ( M/B )) is defined to be the Zariski closure of ( R ( M/B ) o × W R ( M/B ) o )in R ( M/B ) × R ( M/B ).When B = {∗} , D. Davy [Dav] proved that this definition is equivalent to Malgrange’s originaldefinition. In particular Malgrange shows in [Mal01] that there exists a Zariski open subset M o of M such that the restriction of M al ( X ) to Aut ( M o ) is a subgroupoid.2.6.2. Galois groupoid of a rational dominant map.
A rational functions H ∈ C ( R ( M/B )) suchthat H ◦ R Φ = H is a (fibered) differential invariant of Φ. Let Inv (Φ) ⊂ C ( R ( M/B )) be thesubfield of differential invariants of Φ.Let W be a model for Inv and π : R ( M/B ) W be the dominant map induced by theinclusion Inv ⊂ C ( R ( M/B )).
Definition 3.
The (fibered) Galois groupoid is
M al (Φ /B ) = R ( M/B ) × W R ( M/B ) ⊂ Aut ( M/B )Here again, to defined properly this fiber product one needs to restrict π : ( R ( M/S )) o → W on its domain of definition then ( R ( M/B ) × W R ( M/B )) is defined to be the Zariski closure of( R ( M/B ) o × W R ( M/B ) o ) in R ( M/B ) × R ( M/B ).The following lemmas are important in the proof of the specialisation theorem.
Lemma 4.
Let O ⊂ R ( M/B ) o be a Zariski open subset then O × W O = R ( M/B ) o × W R ( M/B ) o . Lemma 5.
Galois groupoid of Φ is the Zariski closure of the set of Taylor expansions of iteratesof Φ . ALOIS GROUPOID AND CONFLUENCE OF DIFFERENCE EQUATIONS 6
Proof. –
Let k be an integer and consider the order k frame bundle R k ( M/B ). From [AC08],the dominant map π : R k ( M/B ) W k induced by the inclusion Inv k ⊂ C ( R k ( M/B )) as thefollowing property : for a general j k ( r ) ∈ R k ( M/B ) the Zariski closure of the fiber π − ( π ( j k ( r )))is the Zariski closure of the R k (Φ) orbit of j k ( r ).Let q : R k ( M/B ) × R k ( M/B ) → Aut k ( M/B ) be the quotient by the diagonal action of Γ. Theimage q ( j k (Φ ◦ r ) , j k ( r )) is j k (Φ) : the order k Taylor expansion of Φ at r (0). Then M al k (Φ) = q ( R k ( M/B ) × W k R k ( M/B )) contains Taylor expansions of Φ.For general m ∈ M , one can find a frame r with r (0) = m such that π − ( π ( j k ( r ))) × { r } ⊂ R k ( M/B ) × { r } is the Zariski closure of the orbit of { r } × { r } for Φ acting on the first factor. Theprojection q ( π − ( π ( j k ( r )))) is the Zariski closure of the set of the Taylor expansions of Φ ◦ n at m .It is also the subset of M al k (Φ /B ) of order k jet of maps with source at m .Let T k be the Zariski closure of the set of all the Taylor expansions of iterates of Φ. Thesubvarieties T k and M al k (Φ) coincide for source out of a closed subvariety of M . By minimality M al k (Φ) ⊂ T k . For n ∈ N let j k Φ ◦ n : M → Aut k ( M/B ) be the map sendind m to the Taylorexpansion of Φ ◦ n at m . This map is rational on M and belongs to M al k (Φ /B ) for general valuesof m thus for any m ∈ M (where the map is defined). This proves the equality. (cid:3) Remark 6.
When Φ is the map arising from a linear difference equation on a vector bundle M → B : (1) the pseudogroup M al (Φ /B ) is “the analog” of the intrinsec Galois group of the equationover the difference field ( C ( B ) , σ )(2) the pseudogroup M al (Φ / ∗ ) is “the analog” of the intrinsec differential Galois group of thedifference equation over C ( B ) with difference operator σ and differential structure given bythe exterior differential d : C ( B ) → C ( B ) ⊗ C [ B ] Ω B A specialisation theorem and a confluence theorem
The specialisation theorem.
When the equation depends on parameters, the base B isfibered on S the parameter space and the map σ : B B preserves the fibers of this map. Thefirst result is a comparison theorem of Galois groupoid of Φ with Galois groupoid of its restrictionon a fiber Φ s : M s M s for s ∈ S Theorem 7.
For all k ∈ N , dim M al k (Φ s /B s ) ≤ dim S M al k (Φ /B ) with equality for general s ∈ S Proof. –
The dimension of
M al k (Φ /B ) is the dimension of M plus the dimension of the Zariskiclosure of a general orbit of R k (Φ) in R k ( M/B ). One has to prove that the dimension of the Zariskiclosure of a particular orbit is smaller that the general one. The proof is done in [AC08]. It canbe adapted from [Bon06] where this inequality is proved for leaves of algebraic foliations.For j k r ∈ R k ( M/B ), V ( j k r ) is the Zariski closure of the orbit of j k r . We will compare Hilbertpolynomial of these subvarieties and deduce the wanted inequalities of dimensions.The ideal of V ( j k r ) is I ( j k r ) = { f ∈ C [ R k ( M/B )] |∀ n ∈ N j k ( r ) ∈ { f ◦ Φ ◦ n = 0 }} and thesubvector space of equation of degree less than or equal to d is also described by these linearequations in f depending on r . Let h j k r ( d ) be the dimension of this space. There exists a Zariskiopen subset O ⊂ R k ( M/B ) and an integer h k such that for any r , h j k r ( d ) ≥ h k with equality for j k r ∈ O .This implies that the Hilbert polynomial of V ( j k r ) is contant for j k r general in R k ( M/B ), andthat for other frames, it is smaller. The dimension of a subvariety is the degree of its Hilbertpolynomial. The theorem is proved. (cid:3)
The confluence theorem.
In this section, we will assume the parameter space to be 1-dimensional with coordinate s . This theorem compares Galois groupoids of Φ with Galois groupoidof X a vector field on M s such that Φ s = Id + ( s − s ) X + o ( s − s ) Theorem 8.
For all k ∈ N , dim M al k ( X/B s ) ≤ dim S M al k (Φ /B ) ALOIS GROUPOID AND CONFLUENCE OF DIFFERENCE EQUATIONS 7
Proof. –
First, the restrictions of differential invariants of Φ at s = s give differential invariantsof X . By expansion around s , R Φ s = Id + ( s − s ) RX + o ( s − s ) and H = H + o ( s − s ) then H ◦ R Φ s = H implies RX · H = 0. This means that the fiber of M al k (Φ /B ) at s contains M al k ( X/B s ) or equivalently Inv ( R k Φ) | s contains Inv ( R k X ).To prove the inequality one has to prove that dim M al k (Φ /B ) s ≤ dim S M al k (Φ /B ). As thecodimension of these varieties are given by the transcendence degree of theirs fields of differentialinvariant, we will compare Inv ( R k Φ) s and Inv ( R k Φ).Let H , . . . H p be a transcendence basis of Inv ( R k Φ) over C ( s ). Because s is a differentialinvariant, one can assume that the restriction of the basis to s = s are well defined rationalfunctions H , . . . , H p and the vanishing order ℓ of dH ∧ . . . ∧ dH p at s = s is minimal. If ℓ = 0,these restrictions are functionally independant, the theorem is proved. If not, let P ( H , . . . , H p ) bethe minimal polynomial of H p over C [ H , . . . , H p − ]. The invariant P ( H , . . . , H p ) can be writtenas ( s − s ) F with F a function of s, H , . . . , H p . Up to some factorisation by s − s one can assumethat there exist a index i such that ( ∂ i P ) is well defined and not zero. One has, up to some signand mod ds , d ( P ( H , . . . H p )) ∧ dH ∧ . . . d dH i . . . ∧ dH p = ∂ i P dH ∧ . . . ∧ dH p = ( s − s ) dF ∧ dH ∧ . . . d dH i . . . ∧ dH p This contradicts the minimality of ℓ . (cid:3) Corollaries about Galois groupoids of discrete Painlev´e equations.
A discrete Pain-lev´e equation is a birational map Φ of C × C fibered above a automorphism without periodicpoints of the curve of the independent variable n . This curve may be non compact. Formallythe difference equation of invariant curves paramaterized by n is the difference equation. Twoproperties are imposed to be called “discrete-Painlev´e equation” :(1) It must have the singularity confinement property. There exists a fibers compactification M of C × C → C such that Φ can be extended as a biholomorphism of M .(2) It must degenerate on a differential Painlev´e equation. There exists a deformation f M of M above C and a deformation e Φ of Φ such that for ǫ ∈ C ∗ ( f M ǫ , e Φ ǫ ) is birrational to ( M, Φ)and at 0 one has the Taylor expansion e Φ ǫ = Id + ǫX + o ( ǫ ) with X a vector field on f M whose trajectories parameterized by open set in C are Painlev´e transcendents. This vectorfield is called a continuous limit of Φ.The classification of the phase spaces of these equations can be found in [Sak01]. The equationsthemselves can be found in [KNY17]. The spaces are classified by families indexes by affine Weylgroups W , each family depends on finite number of parameters a . Then Painlev´e equations aredenoted by dP W ( a ), dP ∗ W ( a ), dP altW ( a ), qP W ( a ) . . . In this article any of these equations will bedenoted by dP W ( a ). In Sakai’s classification [Sak01], it is proved that there exists an invariantrelative 2-form ω ∈ Ω M/ C . In the framework of this article this means that we have an inclusion : M al ( dP W ( a ) / C ) ⊂ { ϕ : ( M n , p ) → ( M n ′ , q ) | ( n, n ′ ) ∈ C , ϕ ∗ ω = ω } . This inclusion, the second property above and Cartan generic local classification of algebraic pseu-dogroups [Car08] enable us to compute Galois groupoid of discrete Painlev´e equations.
Theorem 9.
For any affine Weyl group W and for general values of parameters aM al ( dP W ( a ) / C ) = { ϕ : ( M n , p ) → ( M n ′ , q ) | ( n, n ′ ) ∈ C , ϕ ∗ ω = ω } . Proof. –
First one will determined the codimension 2 subvariety given for n ∈ C by M n,n = M al ( dP W ( a ) / C ) ∩ { ϕ : ( M n , p ) → ( M n , q ) , } . It is a subgroupoid of Inv ( ω n ) = { ϕ : ( M n , p ) → ( M n , q ) | ϕ ∗ ω = ω } . The restriction of these pseudogroups on a fiber M n give algebraic pseudogroupson two dimensional manifold. Such objects have been classified by Elie Cartan [Car08] up toanalytic change of coordinates near a generic point. From the specialisation theorem we know thatdim( M n,n ) k , the algebraic variety of jet of order k of element of M n,n , has a quadratic growth in k . From Cartan results we know that there is only one quadratic growth subgroupoid of Inv ( ω n ): it is Inv ( ω n ) itself.Now the two groupoids dominate C × C and have same fibers above the diagonal, they areequals. (cid:3) ALOIS GROUPOID AND CONFLUENCE OF DIFFERENCE EQUATIONS 8
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ALOIS GROUPOID AND CONFLUENCE OF DIFFERENCE EQUATIONS 9
Guy Casale, Univ Rennes, CNRS, IRMAR-UMR 6625, F-35000 Rennes, France
E-mail address : [email protected] Damien Davy, Univ Rennes, CNRS, IRMAR-UMR 6625, F-35000 Rennes, France
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