Several results on the almost good reduction property to the q-discrete Painlevé equations over the field of p-adic numbers
Abstract
We report on the recent results in the theory of discrete integrable equations reduced modulo a prime number p based on the previous results. This article mainly concerns the q-discrete analogues of the Painlevé equations. We first give a quick review of the idea of `almost good reduction', which is a property that most integrable systems over a finite field Fp possess when reduced from those defined over the field of p-adic numbers Qp. This idea has been introduced in our previous papers and the discrete and q-discrete Painlevé II equations have been proved to have `almost good reduction', which can be interpreted as an arithmetic analogue of the singularity confinement test. In this article we prove that this property holds for several other q-discrete Painlevé equations. The theory of Diophantine integrability is reviewed and its relation to the integrability detection by the algebraic entropy is discussed.
Related
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BibTeX
@misc{kanki2026several,
title = {{Several results on the almost good reduction property to the $q$-discrete Painlevé equations over the field of $p$-adic numbers}},
author = {Masataka Kanki},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33105},
eprint = {2609.33105},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33105},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Several results on the almost good reduction property to the q-discrete Painlevé equations over the field of p-adic numbers
AU - Masataka Kanki
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33105
UR - https://arxiv.org/abs/2609.33105
AB - We report on the recent results in the theory of discrete integrable equations reduced modulo a prime number p based on the previous results. This article mainly concerns the q-discrete analogues of the Painlevé equations. We first give a quick review of the idea of `almost good reduction', which is a property that most integrable systems over a finite field F_p possess when reduced from those defined over the field of p-adic numbers Q_p. This idea has been introduced in our previous papers and the discrete and q-discrete Painlevé II equations have been proved to have `almost good reduction', which can be interpreted as an arithmetic analogue of the singularity confinement test. In this article we prove that this property holds for several other q-discrete Painlevé equations. The theory of Diophantine integrability is reviewed and its relation to the integrability detection by the algebraic entropy is discussed.
ER -