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Galois theory and walks in the quarter plane
Post-edited
Authors : Hardouin, Charlotte (Author of the conference)
CIRM (Publisher )
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)
CIRM (Publisher )
Abstract :
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.
MSC Codes : 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)
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 06/06/2018 Conference Date : 30/05/2018 Subseries : Research talks arXiv category : Combinatorics ; Number Theory Mathematical Area(s) : Combinatorics ; Number Theory Format : MP4 (.mp4) - HD Video Time : 00:49:31 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2018-05-30_Hardouin.mp4 
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Information on the Event
Event Title : Algebra, arithmetic and combinatorics of differential and difference equations / Algèbre, arithmétique et combinatoire des équations différentielles et aux différencesEvent Organizers : Adamczewski, Boris ; Delaygue, E. ; Raschel, Kilian ; Roques, Julien Dates : 28/05/2018 - 01/06/2018 Event Year : 2018 Event URL : https://conferences.cirm-math.fr/1761.html
Citation Data
DOI : 10.24350/CIRM.V.19409503Cite 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 |
See Also
- [Multi angle] The classification of excursions / Author of the conference Mishna, Marni.
- [Multi angle] Gamma functions, monodromy and Apéry constants / Author of the conference Vlasenko, Masha.
- [Multi angle] A supercongruence and hypergeometric motives / Author of the conference Beukers, Frits.
- [Multi angle] Around Jouanolou-type theorems / Author of the conference Moosa, Rahim.
Bibliography
- Dreyfus, T., Hardouin, C., Roques, J., & Singer, M.F. (2018). On the nature of the generating series of walks in the quarter plane. Inventiones mathematicae - https://doi.org/10.1007/s00222-018-0787-z
- Dreyfus, T., Hardouin, C., Roques, J., & Singer, M.F. (2017). Walks in the quarter plane, genus zero case.
