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H 2 Galois theory and walks in the quarter plane

Auteurs : Hardouin, Charlotte (Auteur de la Conférence)
CIRM (Editeur )

 Loading the player... differential relations for special functions Galois theoretic approach differential transcendence criteria walks in the quarter plane geometric uniformisation functional equation for genus zero walks functional equation genus one telescoper criteria for differential transcendence orbit residue genus zero orbit residue genus one ultrametric framework questions from the audience

Résumé : In the recent years, the nature of the generating series of walks in the quarter plane has attracted the attention of many authors in combinatorics and probability. The main questions are: are they algebraic, holonomic (solutions of linear differential equations) or at least hyperalgebraic (solutions of algebraic differential equations)? In this talk, we will show how the nature of the generating function can be approached via the study of a discrete functional equation over a curve E, of genus zero or one. In the first case, the functional equation corresponds to a so called q-difference equation and all the related generating series are differentially transcendental. For the genus one case, the dynamic of the functional equation corresponds to the addition by a given point P of the elliptic curve E. In that situation, one can relate the nature of the generating series to the fact that the point P is of torsion or not.

Codes MSC :
05A15 - Exact enumeration problems, generating functions
12F10 - Separable extensions, Galois theory
12H05 - Differential algebra
12H10 - Difference algebra
30D05 - Functional equations in the complex domain, iteration and composition of analytic functions
39A13 - Difference equations, scaling ($q$-differences)

 Informations sur la Vidéo Réalisateur : Hennenfent, Guillaume Langue : Anglais Date de publication : 06/06/2018 Date de captation : 30/05/2018 Collection : Research talks ; Combinatorics ; Number Theory Format : MP4 (.mp4) - HD Durée : 00:49:31 Domaine : Combinatorics ; Number Theory Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2018-05-30_Hardouin.mp4 Informations sur la rencontre Nom de la rencontre : Algebra, arithmetic and combinatorics of differential and difference equations / Algèbre, arithmétique et combinatoire des équations différentielles et aux différencesOrganisateurs de la rencontre : Adamczewski, Boris ; Delaygue, E. ; Raschel, Kilian ; Roques, JulienDates : 28/05/2018 - 01/06/2018 Année de la rencontre : 2018 URL Congrès : https://conferences.cirm-math.fr/1761.html Citation Data DOI : 10.24350/CIRM.V.19409503 Cite this video as: Hardouin, Charlotte (2018). Galois theory and walks in the quarter plane. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19409503 URI : http://dx.doi.org/10.24350/CIRM.V.19409503

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