Laboratoire Paul Painlevé (UMR - CNRS/USTL 8524)


Aspects algébriques et analytiques des équations aux (q-)différences.   Liens avec Galois classique et arithmétique


13 - 15 Septembre 2004, Lille

Organisateurs : Anne Duval    Changgui Zhang

Programme


Les conférences ont lieu dans la salle de Réunion, 1er étage du bâtiment M2

Lundi 13 septembre

9h30 - 10h : Accueil

10h - 11h : J. Sauloy Algebraic construction of the Stokes sheaf for irregular $q$-difference equations.

11h - 11h30 : Pause café

11h30 - 12h30 : M. van der Put Some algebraic questions concerning $q$-difference equations.

12h30 - 14h30 : Pause déjeuner

14h30 - 15h30 : J. Ecalle The perinomal landscape: equations, functions, numbers.

15h45 - 16h45 : F. Chyzak Taming Apparent Singularities via Ore Closure.

16h45 - 17h15 : Pause café

17h15 - 18h15 : D. Sauzin Resurgence of parabolic curves in C2. 		Mardi 14 septembre

9h - 10h : B. Malgrange Exercices de théorie de Galois différentielle.

10h15 - 11h15 : D. Bertrand Exercices sur les lemmes de zeros.

11h15 - 11h45 : Pause café

11h45 - 12h45 : M. Reversat Sur les extensions abéliennes aux différences et différentielles.

12h45 - 14h30 : Pause déjeuner

14h30 - 15h30 : M. Ismail Asymptotics of $q$ Orthogonal Polynomials and a $q$-Airy Function.

15h45 - 16h45 : L. Haine Askey-Wilson type functions, with bound states.

16h45 - 17h15 : Pause café

17h15 - 18h15 : F. Menous Analytic classification for a class of q-difference equations and some generalizations to systems. 		Mercredi 15 septembre

9h00 - 10h00 : P. Cartier Sur le groupoïde de Galois d'une équation différentielle.

10h15 - 11h15 : Y. André Autour de $\Gamma_q$ et des séries d'Eisenstein (sous réserve).

11h15 - 11h45 : Pause café

11h45 - 12h45 : G.K. Immink Some new results on non-linear difference equations.

12h45 - 14h30 : Pause déjeuner

14h30 - 15h30 : J. Sauloy Monodromie pour les equations aux q-differences complexes.

15h45 - 16h45 : Discussions pour la suite du groupe de travail.

16h45 : Pause café

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Y. André (ENS Paris, France) : Autour de $\Gamma_q$ et des séries d'Eisenstein (sous réserve).
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D. Bertrand (Paris VI, France) : Exercices sur les lemmes de zeros.
Soient $S$ un sous-sch\'ema ponctuel de la sph\`ere de Riemann, et $F = (f_1, ..., f_n)$ une application analytique au voisinage de $S$. Un lemme de z\'eros fournit, en fonction de l'entier $D \geq 0$, un majorant du degr\'e de l'intersection de $S$ avec le sch\'ema des z\'eros des restes $R_D(F)$ des approximants de Pad\'e de degr\'e $\leq D$ de $F$. Les exercices en donneront des exemples quand $F$ est solution d'un syst\`eme diff\'erentiel. Motiv\'es par un travail de transcendance d'Amou et Mataha-alo, des $q$-analogues en seront propos\'es comme probl\`eme d'examen.
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P. Cartier (Jussieu et IHES, France) : Sur le groupoïde de Galois d'une équation différentielle.
Nous montrerons d'abord comment reformuler en termes de groupoïdes la théorie de Galois ordinaire . Nous étendrons ces idées au cas des équations différentielles , en définissant de manière intrinsèque leur groupoïde de Galois différentiel , en deux versions : algébrique et topologique . Au moyen de la notion d'algèbre de Picard-Vessiot , nous montrons comment déduire de là UN groupe de Galois différentiel.
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F. Chyzak (INRIA-Rocquencourt, France) : Taming Apparent Singularities via Ore Closure.
We consider linear functional equations expressed in the framework of univariate skew polynomial rings. Operations like the removal of apparent singularities of a differential equation, finding the minimal-order recurrence satisfied by the coefficients of its formal solutions, and the desingularization of recurrences and q-recurrences share a relation with the calculation of polynomial torsion modules in algebras of skew polynomials. Our analysis here is crucially based on the geometric shape of the singular locus of the functional equation under consideration (fixed singularity, infinite orbit, or periodic orbit). In the differential case, we revisit algorithms by Tsai in view of more efficiency; in the case of recurrences, q-recurrences, and Mahler equations, our algorithmic results seem new, and require a more involved machinery. (Joint work with Ph. Dumas, H. Lê, J. Martins, M. Mishna, and B. Salvy.)
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J. Ecalle (Orsay, France) : The perinomal landscape: equations, functions, numbers.
*Perinomal* equations involve the action of Sl_n(Z) on meromorphic functions of n variables. In many respects, they partake of both difference and q-difference equations. They also display self-duality properties akin to those of homogeneous linear differential equations with polynomial coefficients. Thirdly, perinomal functions constitute a ring, with (for any given degree) a finite set of generators --- hence a definite algebraic flavour, strongly reminiscent of the theory of elliptic or automorphic functions. Lastly, and chiefly, perinomal functions, together with the transcendental perinomal numbers attached to them, are the proper tool for understanding *arithmetical dimorphy*, particularly in the ring of multizetas, and for describing the corresponding *irreducibles*.
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L. Haine (Louvain, Belgique) : Askey-Wilson type functions, with bound states.
I will discuss the following problem. To find all doubly infinite tridiagonal matrices for which some family of eigenfunctions is also a family of eigenfunctions of a q-difference equation in the spectral variable. One family of solutions is intimately related to the Askey-Wilson polynomials. Another family arises as special soliton solutions to the Toda lattice equations. There is an exciting connection between this second family of solutions and a "finite number of terms phenomenon" for the fundamental solution of some discrete heat equations.
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G.K. Immink (Groningen, Pays-Bas): Some new results on non-linear difference equations
Formal solutions of difference equations with a level ``1^+'', are, in general, not multi-summable. We consider the case of two levels: 1 and 1^+. In this case it is possible to construct right inverses of a corresponding linear difference operator on Banach spaces of quasi-functions. Using this result one can prove accelero-summability of formal solutions.
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M. Ismail (Orlando (FL), USA) : Asymptotics of $q$ Orthogonal Polynomials and a $q$-Airy Function.
We establish Plancherel-Rotach type type asymptotics for all the polynomials in the Askey scheme with unbounded recurrence coefficients $(q > 1)$. This includes the polynomials $\left\{h_n(x\,|\,q)\right\}$ of Askey, Ismail and Masson, and the $q$-Laguerre polynomials. We also treat the Al-Salam-Chihara, and continuos dual $q$-Hahn polynomials when $q > 1$. A $q$-analogue of the Airy function emerges and is related to a special $q$-Bessel function which was considered by Ramanujan in his lost notebook. We also give a $q$-analogue of the Airy kernel.
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B. Malgrange (Grenoble, France) : Exercices de théorie de Galois différentielle.
j'examinerai des exemples de la situation où l'on a deux feuilletages emboités (intuitivement:on adjoint les solutions d'une première équation,puis d'une seconde) .Dans le cas linéaire,cela conduit à un groupe et un sous-groupe distingué;dans le cas non linéaire général,la situation est bien plus compliquée....
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F. Menous (Orsay, France) :Analytic classification for a class of q-difference equations and some generalizations to systems .
Using the techniques developped by Jean Ecalle for the study of nonlinear differential equations, we prove that the $q$-difference equation \[ x \sigma_q y = y + b ( y, x ) \] with $( \sigma_q f ) ( x ) = f ( q x )$ ($q > 1$) and $b ( 0, 0 ) = \partial_y b ( 0, 0 ) = 0$ is analytically conjugated to one of the following equations : \[ x \sigma_q y = y \] or \[ x \sigma_q y = y + x \] We will also give some generalizations of this result to systems of $q$-difference equations, where problems of resonance and small divisors seem to appear.
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M. van der Put (Groningen, Pays-Bas) : Some algebraic questions concerning $q$-difference equations.
In the first part we investigate Krichever difference modules of rank one in comparison with the better known Drinfeld modules of the same rank. The second question is the descent problem for $q$-difference equations. Good moduli space for meromorphic $q$-difference equations do exist and this solves the descent problem.
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M. Reversat (Toulouse, France) : Sur les extensions abéliennes aux différences et différentielles .
Soit $\mathcal C$ une courbe définie sur le corps $\mathbb C$, projective, connexe et lisse, de genre $g>0$. Soient $\sigma $ un automorphisme (resp. une dérivation) de $\mathcal C$ et $\mathfrak K$ le corps des fonctions rationnelles sur $\mathcal C$. On expose les grandes lignes d'une description des extensions abéliennes de $(\mathfrak K, \sigma )$ à l'aide des $\mathcal D$-modules sur la jacobienne de la courbe $\mathcal C$.
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J. Sauloy (Toulouse, France) :Algebraic construction of the Stokes sheaf for irregular $q$-difference equations.
The local analytic classification of irregular linear $q$-difference equations has recently been obtained by J.-P. Ramis, J. Sauloy and C. Zhang. Their description involves a $q$-analog of the Stokes sheaf and theorems of Malgrange-Sibuya type and is based on a discrete summation process due to C. Zhang. We show here another road to some of these results by algebraic means and we describe the $q$-Gevrey devissage of the $q$-Stokes sheaf by holomorphic vector bundles over an elliptic curve.
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J. Sauloy (Toulouse, France) : Monodromie pour les equations aux q-differences complexes.
Pour les equations aux q-differences fuchsiennes, il est devenu traditionnel de considerer que ce qui tient lieu de monodromie est code par la matrice de connexion de Birkhoff. Je discuterai de la monodromie locale en $0$ et en $\infty$ et de sa description par des fibres plats, puis de l'extension de cette description au cas irregulier (phenomene de Stokes): jusque la, il s'agit de resultats bien etablis (Ramis-Sauloy-Zhang). Puis je tenterai de montrer comment la monodromie "globale", codee par la matrice de connexion de Birkhoff etendue au cas irregulier, peut etre "localisee" en les singularites intermediaires
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D. Sauzin (Paris, France) : Resurgence of parabolic curves in C2.
A parabolic germ of analytic diffeomorphism of $(C2,0)$ always admits invariant curves which may be called stable and unstable manifolds. We show, under some non-degeneracy hypothesis, how two such curves can be obtained as Borel sums of a single formal series in~$z^{-1}$ and~$z^{-1}log z$, where~$z$ is a large variable. The formal series is obtained as solution of a nonlinear difference equation of the form $x(z+1)-2x(z)+x(z-1)=f(x(z),x(z)-x(z-1))$. Generically, this series is divergent and the two sums are not the analytic continuation one of the other; the leading order of their difference, which is exponentially small, is determined by a pair of complex constants. We analyse this phenomenon in the framework of Resurgence theory and prove that these constants depend analytically on parameters. They vanish for the time-1 map of an analytic vector field but not for a generic map.
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