On Drach s Conjecture

Transcription

1 On Drach s Conjecture Guy Casale Contents The Galois groupoid of a foliation 2. D-variety over L D-Lie groupoid over L D-Lie algebras over L, first part Brackets and differential Spencer bracket Fiberwise bracket D-Lie algebra over L, second part The Galois groupoid Godbillon-Vey sequences and Drach s resolvante 6 3 Isomonodromic deformation 6 4 The counter example of R. Garnier 7 5 Computations of examples 8 5. The dim trans Gal (E(λ)) = The dim trans Gal (E(λ)) = The dim trans Gal (E(λ)) = The dim trans Gal (E(λ)) = Dicritical singularities? 9 Introduction In numerous papers between 898 and 920 ([5, 6]) J.Drach gave a conjectural characterization of the sixth Painlevé transcendents using the nonlinear Galois group (or pseudogroup) of an auxilary differential equation. Because of gaps in his definition ([4]) this galoisian theory was almost forgotten except by E. Vessiot and J-F.Pommaret. Recentlty Two definition of the Galois pseudogroup of a differential equation were given by H.Umemura and B.Malgrange. This two definitions are similar to the one proposed by Vessiot to correct Drach s definition ([]). In [0], H.Umemura gives a precise statement of the galoisian characterization of the sixth Painlevé transcendents by mean of his infinitesimal Galois group. In this paper he remarks that a part of this conjecture can be easily proved using Malgrange s Galois groupoid. The author is supported by JSPS Fellowship for Foreign Researchers (FY2004) and EIF Marie-Curie Fellowship Address: Departament de matematiques, Universitat Autonoma de Barcelona, 0893 Bellaterra (Barcelona) Espana address: casale@mat.uab.es & casale@picard.ups-tlse.fr

2 When the equations of the Galois groupoid are written down, it becomes clear that Malgrange s proof is just formulae of Schlesinger equations for isomonodromic deformation. In this note, we will emphase the Garnier system point of view in order to give a counter-example to the P -version of the conjecture. An equation with a Galois groupoid smaller than its automorphism groupoid is called special. The Drach s conjectures are the following and (A) The equation dy = x λ dx x(x ) only if λ is a solution of P 6. y(y ) y λ is special if and (B) The equation dy = is special if and only if λ dx 2 y λ is a solution of P or P 2. In a first part we briefly introduce the Galois groupoid of a foliation over a (partial) differential field L. The Galois groupoid of a differential equation over a (ordinary) differential field K, E : y (n+) = E(y, y...y (n) ) K(y,..., y (n) ) is the Galois groupoid of the associated foliation over L = ( K(y,..., y (n) ), x,... y (n)) where the y s are now constants for x and x K =. This is the Galois groupoid of E over K. When there is no explicit ground field, we assume that K is the differential field generated by the coefficients of E. This is the case in the statement of conjectures (A) and (B). In a second part we applied results obtained in [2, 3] about the Galois groupoid of an order one equation to construct a isomonodromy preserving deformation from the equation. Then the IF part of the conjectures appears to be computation done by R.Fuchs [7] and generalized by R.Garnier [8]. This point of view gives the counter-example to the (B) conjecture explained in the third part. This counter-example was already in the article of Garnier. In a third part, we calculate the Galois groupoid of the (A) equation depending on transcendance of λ in few simpler cases. In a fourth part, these calculations will be applied to determine the simplest dicritical singularities of (C 2, 0) with some transversal geometric structure (discret, euclidiens or affine), (projective?? unknown; it is (A) conjecture) Introduction The Galois groupoid of a foliation Let L be a differential field with derivations,..., n. The field of constants of L is supposed algebraically closed and its characteritic is zero. In this article we assume this field is C. T(L) is the L-vector space generated by the s and T (L) its dual over L with dual base d,...,d n. There is two (dual) ways to define a foliation over L (or a foliation on spec(l)). The first one is to give the equations of the involutive distribution i.e. a subspace of T (L) stable by the exterior differential d : T (L) Λ 2 T (L). The second one is to give the solutions of these equations over L i.e. a subspace of T(L) stable by the Lie bracket. In this article, a foliation must be thought as a particular D-Lie algebra. As general D-Lie algebras can have no solution over L, we will emphase the former point of view.. D-variety over L Let A m L be the affine space of dimension m over L with coordinates z,...,z m. The space of order q jets of section of J q (A m L /spec L ) or J q(a m L /L) is the variety defined by the ring L [z α i i m, α N n, α k]. 2

3 The space of jets of sections J(A m L /L ) = limj q(a m L /L) is a scheme of countable type over L. The derivations of L act on its ring by the following formulae D i : O J k O J k+, i n D i = i + j,α z α+ɛ i j z α j. Definition. An affine D-variety over L is a sub-scheme Z of J(A m L /L) defined by a differential ideal (i.e. a ideal stable under the action of the D i s). Example.2 Let Z be an affine variety over L. Its ideal I generates a differential ideal I of O J(A m L /L ). This D-variety is the space of section of Z over L: J(Z /L ). Definition.3 Let Z be a D-variety over L and L E be a extension of differential fields. A E-point of Z is a morphism O Z E over the inclusion. A differential point is given by a differential morphism. Remark.4 A E-point is a section of Z over E but a D-E-point is the jet of a section of Z 0 over E. For V is a L-vector space, the set V /L = {V /L (E) E is a differential extension of L} is the sheaf of local sections of V /L..2 D-Lie groupoid over L This notion is an algebraic formulation of the basic following example. Example.5 Let α be a rational -form on C n. The set of formal transformations of C n : ϕ : Ĉ n, a Ĉ n, b leaving α invariant is a groupoid acting on C n defined by rational differential equations Let s build the space of order k jets J (spec L spec L) (J for short) for a differential field. Let L () et L (2) be two copies of L and O J k be the ring [ ] L () L (2) zi α, C det(z ɛ j i ) ; i n, α Nn, α k. The spaces Jk = spec O Jk are groupoids on spec L. This groupoid structure is given by the following maps. The projections s (for source) on spec L () and t (for target) on spec L (2) are self-explained. The composition c : (Jk, s) L (Jk, t) J k is defined on the order 0 jets by the projection of the product spec L C spec L C spec L on the first and third factors. On higher order jets it is defined by using the formulae for composition of formal power series. The inversion is defined by the inversion formulae for formal power series. These maps satisfy some commutative diagrams [?] (obvious in the framework of jets space) and the spaces Jk are groupoids over L. Moreover, one gets derivations D i : O J k O J k+, i n by D i = () i + j z ɛ i j (2) j + j,α z α+ɛ i j z α j We set J = lim Jk et O J = lim O J. This space is a scheme of infinite (countable) type with a k structure of groupoid over L and a structure of D-variety over L (i.e. a lift of the action of T(L) from L to O J ) 3

4 Definition.6 A D-Lie groupoid over L is a subgroupoid of J defined by a perfect differential ideal. Remark.7 The hypothesis perfect is not relevant. By a theorem of B.Malgrange, every non-reduced D-Lie groupoid is in fact reduced. This is proved in the analytic framework in [9]. Example.8 The D-Lie groupoid defined by the ideal (0), i.e. J itself, is called the groupoid of point transformations of spec L on C. Sometimes, it will be denoted by Aut(spec L /C ) or Aut(L /C )..3 D-Lie algebras over L, first part As for D-Lie groupoid, let s begin with a basic exemple. Example.9 Let ω be a rational intégrable (i.e. it satisfies Frobenius integrability condition ω dω = 0) -form on C n. The sheaf of local analytic vectors fields X such that ω(x) = 0 is a sheaf in C-Lie algebra. Let T (L) be the L-vector space generated by the differentials of L and ST (L) = L[a,...,a n ] its symetric powers ring. The space T L = spec ST (L) is the tangent space of spec L. For a extension E of L, T L (E) is the space of E-point of T L. The space of order k jets of sections of T L on spec L, J q (spec L T L ) (J q T for short), is defined by the following ring. Let O Jq T = L [a α i ; i n, α N n, 0 α q]. The space of jets of order q of vector fields on spec L is spec O Jk T. The space J T with ring O J T is defined by limits. The ring O J T is a D-variety, the derivation T(L) of L act on it by D i = () i + j,α a α+ɛ i j a α j. It is a D-vector space: the linear stucture is given by the L-vector space of linear partial differential equations L J T O J T. The Lie bracket on the vector fields on spec L with coefficient in E: T L (E) define a Lie bracket on the space J T L (D-E) of differential E-points. Temporary definition A D-Lie algebra over L is a sub-d-vector space L of J T such that the differential points of L are stable under Lie bracket. A less differential definition will be given in the next section following B.Malgrange [9]. This definition will use a prolongation of the Lie bracket on J T called the Spencer bracket and the stability condition will be on the ordinary points (=sections) of the jet space..4 Brackets and differential There is two brackets defined on J q T. The first bracket is defined on J q T and take values in J q T. It is called Spencer (or differential) bracket. It allows us to named J q T a Lie algebroid [?]. The second one is defined on fibers of J q T and take values in J q T. It is called the fiberwise (or algebraic) bracket..4. Spencer bracket The construction of this bracket (denoted by [.,. ]) follows the diagonal method [?, 9]. First, the frame spaces are defined. The space of order q-frame on spec L is R q (spec L) = Jq (Ĉ n, 0 spec L) or R q for short is defined by the ring [ ] O Rq = L ri α, det(r ɛ j i ) ; i n, α Nn, α k. 4

5 This space is a principal homogeneous space on spec L with structural group the linear algebraic group GL q (C n ) = Jq (Ĉn, 0 Ĉ n, 0) action by composition. The next step is to study the composition ρ. Let R q () et R q (2) be two copies of R q ρ : R q () R q (2) Jq defined for couples (r, s) of q-frames by r s and the morphism of ring induced. This map is the quotient by the diagonal action of GL q (C n ) by source composition. The tangent of ρ : Tρ : T(R () q R (2) q ) /R () q TJ q /L () identifies vector fields on R q () R q (2) in the kernel of the first projection and invariant under the action of GL q (C n ) and vector fields on Jq in the kernel of the source. Because the constructions of the vertical tangent and the jet space commute, one have J q T T(J q /L ()) id. From an other side, the identification TR q avec T(R q () R q (2) ) () /R diag is equivariant under the action q of GL q (C n ). From these identifications, ρ : TR q J q T is the quotient by GL q (C n ). Definition.0 The Spencer bracket on sections of J q T is the bracket induced by the Lie bracket on R q. By duality, this bracket gives a differential on the sections of L Jq T..4.2 Fiberwise bracket This bracket (denoted by {.,. } ) is defined by the formulae giving j q [X, Y ] in terms of j q X et j q Y for two vector fields on C n..5 D-Lie algebra over L, second part Definition..6 The Galois groupoid Definition.2 Let F be a foliation over L. The smallest D-Lie groupoid whose D-Lie algebra contain F is the Galois groupoid of F. Let E be an ordinary order n differential equation over an ordinary differential field K: E K(y,...,y (n) ) = L. The field L is a differential field with derivations x,..., y (n) defined by xy (j) = 0, y (i)y (j) = δ j i, and x K =, y (i) K = 0. The equation E define a foliation F E over a algebraic extension of L. Its Galois groupoid will be denoted Gal(E /K ). 5

6 2 Godbillon-Vey sequences and Drach s resolvante Definition 2. A Godbillon-Vey sequence for a -form ω 0 over L is a sequence of -forms ω n such that : n ( ) n dω n = ω k ω n k+. k The length of a sequence is the integer l such that ω n = 0 for all n l. k=0 Let F be the foliation define by ω 0. In our previous article [2], we proved that the Galois groupoid of a codimension one foliation is related to the minimal length of its Godbillon-Vey sequences when it is less or equal three. Using the point of view of [3] and results of [], this characterization can be extend for equation over any differential field. Theorem 2.2 The Galois groupoid of F is smaller than Aut(F) if and only if there exist a length three Godbillon-Vey sequences for ω 0. Length three Godbillon-Vey sequences are also called sl 2 -triplet. When the foliation admits a length three sequence, this sequence is not unique. Nevertheless if such a sequence exists, any couple ω 0, ω satisfying the first equation can be completed by a third form satisfying the two other equations. For an equation, a unique triplet begins with ω 0 = dy A dx et ω = y Ad x. This is the vertical triplet. The second equation implies ω 2 = 2 2Ad + R ω 0 for a R in L and the third equation gives 3 y A + xr + A y R + 2 y A R = 0 This equation appears in the thesis of J.Drach [4] and is called resolvante equation. Remark 2.3 The Riccati equation are always special. For such an equation, y 3 A = 0. A rational solution of the resolvante equation in L is R = 0. 3 Isomonodromic deformation The length 3 Godbillon-Vey sequences can be used to compute some special first integrals of the foliation by the followings three steps : () solve dg = ω 2 Gω + G2 2 ω 0, (2) solve df = ( ω + Gω 0 )F, (3) solve dh = Fω 0. These H s are integrals of the foliation. If we use the special sequence to do this computation, one obtains a first integral such that : { S y H = R x H + A y H = 0 The first equation gives a family of singular projective structure on P. The singularities are the poles of R where the equation has no holomorphic injective solutions (the charts of the projective structure). The singularities can be apparent (there is a holomorphic solution) or not. The second equation says that this family is integrable i.e. the charts can be chosen constant along the trajectories of the vector fields x + A y. Because of the singularities of this vector field and the tangent locus of the trajectories with the vertical fibers (x = cste), this family is not trivial. The Drach resolvante is the integrability condition for this system of p.d.e. 6

7 One can linearize this equation by y H = P 2 and it gives the isomonodromic equations of an order two scalar differential equation studied by R. Fuchs and R. Garnier : { 2 y P = 2 RP x P + A y P = 2 yap A 2 2 system for these equations is given by : ( ) P d = P 2 ( 2 ω 2 ω 0 2 ω 2 2 ω The following problem is a generalisation of Drach s conjecture. ) (P P 2 ) Problem When an order one equation E /K is given, is there a isomonodromic deformation of a rank two system along E with coefficient in K(y)? The (A) conjecture is an answer to this problem for E A = dy = x λ dx (B) conjecture is for dy = dx 2 y λ have some solutions : x(x ) y(y ) y λ and K = C(x)(< λ >). The. From Schlesinger, Fuchs and Garnier we know that these problems when λ is solution of P 6, there exists a R K(y) solution of the resolvante equation for E A when λ is solution of P or P 2 there exists a solution of the resolvante equation for E B Is there some others solution? 4 The counter example of R. Garnier If we assume R = m n= 2 a n (y λ) n, the Drach s resolvante gives a m = 0 a m = ma mλ a i = (i + )a i+ λ i a 2 i+2 a = 4λ a 0 = 2(λ ) 2 a = 2λ a 2 = 3 2 For m 4, this system can be reduced to an order two equation : m = λ = c m = 2 λ = c 2 λ + c m = 3 λ = c 3 (6λ 2 + x) + c 2 λ + c m = 4 λ = c 4 (2λ 3 + λx) + c 3 (6λ 2 + x) + c 2 λ + c for some constants c, c 2, c 3 et c 4. This gives the proof of the IF part of the conjecture. The counterexample of the ONLY IF part is given by the resolvante equation for m = 5. For m 5, the system given by the resolvante equation cannot be reduced. These systems give order m equation satisfied by λ. For m = 5, this equation is ( λ 4 λ ) = 6a 3 λ 2a ( a3 ) λ

8 with a 5 = c 5 a 4 = 5c 5 λ + c 4 a 3 = 0c 5 λ 2 + 4c 4 λ 3c 2 5x + c 3 Theorem 4. (Garnier) This equation has some movable branching points. to be completed give the proof (B) conjecture is false 5 Computations of examples Let E(λ) be the equation x 2 dy x λ(x) dx = y2 y λ(x) over the field C({x}) of germs of meromorphic functions on (C, 0). 5. The dim trans Gal (E(λ)) = 0 This assumption implies that E(λ) admits a first integral in the field C({x})(y). Let H be such an integral, one can assume that H = c(x) Πn i= (y a i(x)) Π n i= (y b i(x)) with a i s and b i s algebraic over C({x}). Proposition 5. The function H is a first integral of E(λ) if and only if c = 0 and (a (x),...,a n (x), b (x),...,b n (x)) is a germ of trajectory in C 2n+ of X = x2 x λ x + a 2 i + b 2 i a i λ a i b i λ b i where λ is defined by λ a i λ b i = 0. In this case λ(x) = λ(a (x),...,a n (x), b (x),...,b n (x)) Proof. This is just a calculation. Let s begin by the if part. 0 = E.H H = x2 xh + y2 yh x λ(x) H y λ(x) H = x2 xh + y2 yh x λ(x) [ H y λ(x) H = x2 x λ(x) c c a i y a i + b i y b i ] [ + y2 y λ(x) y a i ] b i. Because the residue in y = a i and y = b i must be zero, one have that the a i (x) s and b i (x) s are solutions of E(λ). Because its value in y = 0 must be zero, c = 0. Moreover yh = so H y a i (x) y b i (x) λ = λ(a (x),...,a n (x), b (x),...,b n (x)). The only if part is just the converse calculation. Theorem 5.2 If E(λ) admits a first integral rational in y then the a i s, b i s and λ are algebraic functions of x and the first integral is in fact rational in x and y. 8

9 Proof. In order to prove this theorem, we have to show that the trajectories of X are algebraic. For this we will compute 2n algebraic first integrals. Let h,...,h 2n 3 be the n 3 roots of Q(y) = y a i y b i different from λ and H j = Πn i= (h j a i ). From dh j Π n i= (h j b i ) H j = da i h j a i db i h j b i, one gets that these functions are not constants and that XH j H j = a 2 i a i λ h j a i b 2 i b i λ h j b i = λ 2 + h2 i h j λ a i λ h i λ h j a i λ 2 h j λ = 0 + h2 j b i λ h j λ h j b i Almost the same computation proves that K = Πn i= ( a i) Π n i= ( b i), K = Πn i= ( a i) Π n i= ( b i) and K x = Πn i= (x a i) Π n i= (x b i) are also first integrals of X. Proof of the independence of the first integrals Proof in the case h i not all different 5.2 The dim trans Gal (E(λ)) = 5.3 The dim trans Gal (E(λ)) = The dim trans Gal (E(λ)) = 3 No solution known here: it is the (A) conjecture! 6 Dicritical singularities? References [] Casale, G. - Suite de Godbillon-Vey et intégrale première, C. R. Acad. Sci. Paris 335 (2002) [2] Casale, G. - Feuilletages de codimension un, groupoïde de Galois et intégrales premières, to appear in Ann. Inst. Fourier (2006) [3] Casale, G. - Le groupoïde de Galois de P et son irréductibilité, submitted (2006) [4] Drach, J. - Essai sur une théorie générale de l intégration et sur la classification des transcendantes, Ann. Sci. Écoles Normale Sup. 5 (898) [5] Drach, J. - Sur le groupe de rationalité des équations du second ordre de M. Painlevé, Bull. Sci. Math. 39 (95) [6] Drach, J. - L équation différentielle de la balistique extérieur et son intégration par quadratures, Ann. Sci. École Normale Sup. 37 (920) 94 [7] Fuchs, R. - Über Linear homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singulären Stellen, Math. Ann. 63 (907) [8] Garnier, R. - Sur les équations diffŕerentielles du troisième ordre dont l intégrale générale est uniforme et sur une classe d équations nouvelles d ordre supérieur dont l intégrale générale a ses points critiques fixes, Ann. Sci. École Normale Sup. 29 (92) 26 [9] Malgrange, B. - Le groupoïde de Galois d un feuilletage, Monographie 38 vol 2 de L enseignement mathématique (200)

10 [0] Umemura, H. - Monodromy preserving deformation and differential Galois group I, Analyse complexe, systèmes dynamiques, sommabilité des séries divergentes et théories galoisiennes I, Astérisque ndeg296 (2004) [] Vessiot, E. - Sur la réductibilité et l intégration des systèmes complets, Ann. Sci. École Normale Sup. 29 (92)