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arXiv · 2008.11101

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

Ruichao Jiang·Javad Tavakoli·Yiqiang Zhao
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Abstract

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

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Ruichao Jiang, Javad Tavakoli, Yiqiang Zhao. 2020-08-19. An upper bound and criteria for the Galois group of weighted walks with rational coefficients in the quarter plane. https://arxiv.org/abs/2008.11101

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