Abstract
The kernel method is an essential tool for the study of generating series of walks in the quarter plane. This method involves equating to zero a certain polynomial - the kernel polynomial - and using properties of the curve - the kernel curve - this defines. In the present paper, we investigate the basic properties of the kernel curve (irreducibility, singularities, genus, uniformization, etc.).
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under the Grant Agreement No 648132. The authors would like to thank ANR-19-CE40-0018, ANR-13-JS01-0002-01, ANR-11-LABX-0040-CIMI within the program ANR-11-IDEX-0002-0, LabEx PERSYVAL-Lab ANR-11-LABX-0025-01, the Simons Foundation (#349357, Michael Singer).
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Notes
- 1.
In several papers it is not assumed that \(\sum _{i,j} d_{i,j}=1\). But after a rescaling of the t variable, we may always reduce to the case \(\sum _{i,j} d_{i,j}=1\).
- 2.
The maple worksheet is available at https://singer.math.ncsu.edu/ms_papers.html.
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The authors want to warmly thank the referees for their detailed and helpful comments.
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Dreyfus, T., Hardouin, C., Roques, J., Singer, M.F. (2021). On the Kernel Curves Associated with Walks in the Quarter Plane. In: Bostan, A., Raschel, K. (eds) Transcendence in Algebra, Combinatorics, Geometry and Number Theory. TRANS 2019. Springer Proceedings in Mathematics & Statistics, vol 373. Springer, Cham. https://doi.org/10.1007/978-3-030-84304-5_3
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