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

@article{CRMATH_2021__359_5_563_0,
     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},
}

TY  - JOUR
AU  - Ruichao Jiang
AU  - Javad Tavakoli
AU  - Yiqiang Zhao
TI  - An upper bound and finiteness criteria for the Galois group of weighted walks with rational coefficients in the quarter plane
JO  - Comptes Rendus. Mathématique
PY  - 2021
SP  - 563
EP  - 576
VL  - 359
IS  - 5
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.196
LA  - en
ID  - CRMATH_2021__359_5_563_0
ER  -

%0 Journal Article
%A Ruichao Jiang
%A Javad Tavakoli
%A Yiqiang Zhao
%T An upper bound and finiteness criteria for the Galois group of weighted walks with rational coefficients in the quarter plane
%J Comptes Rendus. Mathématique
%D 2021
%P 563-576
%V 359
%N 5
%I Académie des sciences, Paris
%R 10.5802/crmath.196
%G en
%F CRMATH_2021__359_5_563_0

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/

Bibliographie
Cité par

[1] Gunnar Billing; Kurt Mahler On exceptional points on cubic curves, J. Lond. Math. Soc., Volume s1-15 (1940) no. 1, pp. 32-43 | DOI | MR | Zbl

[2] Mireille Bousquet-Mélou; Marni Mishna Walks with small steps in the quarter plane, Algorithmic probability and combinatorics (Contemporary Mathematics), Volume 520, American Mathematical Society, 2010, pp. 1-40 | DOI | MR | Zbl

[3] Thomas Dreyfus; Kilian Raschel Differential transcendence & algebraicity criteria for the series counting weighted quadrant walks, Publ. Math. Besançon, Algèbre Théorie Nombres, Volume 1 (2019), pp. 41-80 | Numdam | Zbl

[4] Johannes J. Duistermaat Discrete integrable systems. QRT maps and elliptic surfaces, Springer Monographs in Mathematics, Springer, 2010 | Zbl

[5] Guy Fayolle; Roudolf Iasnogorodski Random walks in the quarter-plane: Advances in explicit criterions for the finiteness of the associated group in the genus 1 case, Markov Process. Relat. Fields, Volume 21 (2015) no. 4, pp. 1005-1032 | MR

[6] Guy Fayolle; Roudolf Iasnogorodski; Vadim Malyshev Random walks in the quarter plane, Probability Theory and Stochastic Modelling, 40, Springer, 2017 | MR | Zbl

[7] Guy Fayolle; Kilian Raschel On the holonomy or algebraicity of generating functions counting lattice walks in the quarter-plane, Markov Process. Relat. Fields, Volume 16 (2010) no. 3, pp. 485-496 | MR | Zbl

[8] Guy Fayolle; Kilian Raschel Random walks in the quarter-plane with zero drift: an explicit criterion for the finiteness of the associated group, Markov Process. Relat. Fields, Volume 17 (2011) no. 4, pp. 619-636 | MR | Zbl

[9] Leopold Flatto Two parallel queues created by arrivals with two demands. II, SIAM J. Appl. Math., Volume 45 (1985) no. 5, pp. 861-878 | DOI | MR | Zbl

[10] Leopold Flatto; S. Hahn Two parallel queues created by arrivals with two demands. I, SIAM J. Appl. Math., Volume 44 (1984) no. 5, pp. 1041-1053 | DOI | MR | Zbl

[11] H. Hopf Vektorfelder in n-dimensionalen Mannifaltigkeiten, Math. Ann., Volume 96 (1927) no. 1, pp. 225-249 | DOI

[12] Manuel Kauers; Rika Yatchak Walks in the quarter plane with multiple steps, Proceedings of FPSAC (Discrete Mathematics & Theoretical Computer Science) (2015), pp. 25-36 | Zbl

[13] Irina Kurkova; Kilian Raschel Explicit expression for the generating function counting Gessel’s walks, Adv. Appl. Math., Volume 47 (2011) no. 3, pp. 414-433 | DOI | MR | Zbl

[14] Vadim Malyshev Positive random walks and Galois theory, Usp. Mat. Nauk, Volume 1 (1971), pp. 227-228 | MR

[15] Barry Mazur Modular curves and the Eisenstein ideal, Publ. Math., Inst. Hautes Étud. Sci., Volume 47 (1977), pp. 33-186 | DOI | Numdam | Zbl

[16] Barry Mazur Rational isogenies of prime degree, Invent. Math., Volume 44 (1978), p. 129 | DOI

[17] Barry Mazur; John T. Tate Points of order 13 on elliptic curves, Invent. Math., Volume 22 (1973) no. 1, pp. 41-49 | DOI | MR | Zbl

[18] Joseph H. Silverman The arithmetic of elliptic curves, Graduate Texts in Mathematics, 106, Springer, 2009 | MR | Zbl

[19] Joseph H. Silverman; John T. Tate Rational points on elliptic curves, Undergraduate Texts in Mathematics, Springer, 2015 | Zbl

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

About a possible analytic approach for walks in the quarter plane with arbitrary big jumps

Guy Fayolle; Kilian Raschel

C. R. Math (2015)