An upper bound and finiteness criteria for the Galois group of weighted walks with rational coefficients in the quarter plane

Ruichao Jiang; Javad Tavakoli; Yiqiang Zhao

doi:10.5802/crmath.196

Combinatoire, Probabilités

An upper bound and finiteness criteria for the Galois group of weighted walks with rational coefficients in the quarter plane
[Un majorant et des critères de finitude pour le groupe de Galois de marches pondérées avec des coefficients rationnels dans le quart de plan]

Ruichao Jiang ¹ ; Javad Tavakoli ¹ ; Yiqiang Zhao ²

¹ The University of British Columbia Okanagan, Kelowna, BC V1V 1V7, Canada
² Carleton University, Ottawa, ON K1S 5B6, Canada

Comptes Rendus. Mathématique, Volume 359 (2021) no. 5, pp. 563-576.

Résumé
Abstract

En utilisant le théorème de Mazur sur les torsions de courbes elliptiques, on obtient un majorant 24 pour l’ordre du groupe fini de Galois $ℋ$ associé aux marches pondérées dans le quart de plan $ℤ_{+}^{2}$ . Le critère explicite pour que $ℋ$ soit d’ordre 4 ou 6 est obtenu par un simple argument géométrique. En utilisant des polynômes de division, un critère récursif pour $ℋ$ d’ordre $4 m$ ou $4 m + 2$ est également obtenu. Comme corollaire, un critère explicite pour que $ℋ$ soit d’ordre 8 est donné et est beaucoup plus simple que la méthode existante.

Using Mazur’s theorem on torsions of elliptic curves, an upper bound 24 for the order of the finite Galois group $ℋ$ associated with weighted walks in the quarter plane $ℤ_{+}^{2}$ is obtained. The explicit criterion for $ℋ$ to have order 4 or 6 is rederived by simple geometric arguments. Using division polynomials, a recursive criterion for $ℋ$ to have order $4 m$ or $4 m + 2$ is also obtained. As a corollary, an explicit criterion for $ℋ$ to have order 8 is given through a method simpler than the existing one.

Reçu le : 2020-09-09
Révisé le : 2021-01-16
Accepté le : 2021-03-03
Publié le : 2021-07-13

DOI : 10.5802/crmath.196

Affiliations des auteurs :

Ruichao Jiang ¹ ; Javad Tavakoli ¹ ; Yiqiang Zhao ²

¹ The University of British Columbia Okanagan, Kelowna, BC V1V 1V7, Canada
² Carleton University, Ottawa, ON K1S 5B6, Canada

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Ruichao Jiang and Javad Tavakoli and Yiqiang Zhao},
     title = {An upper bound and finiteness criteria for the {Galois} group of weighted walks with rational coefficients in the quarter plane},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {563--576},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {5},
     year = {2021},
     doi = {10.5802/crmath.196},
     language = {en},
}

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Ruichao Jiang; Javad Tavakoli; Yiqiang Zhao. An upper bound and finiteness criteria for the Galois group of weighted walks with rational coefficients in the quarter plane. Comptes Rendus. Mathématique, Volume 359 (2021) no. 5, pp. 563-576. doi : 10.5802/crmath.196. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.196/

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