关于$q$-离散Painlevé方程在$p$-adic数域上几乎良好约化性质的若干结果
Several results on the almost good reduction property to the $q$-discrete Painlevé equations over the field of $p$-adic numbers
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中文总结 AI 辅助
本文证明多个$q$-离散Painlevé方程具有几乎良好约化性质，并探讨其与丢番图可积性及代数熵检测的联系。
中文摘要 AI 辅助
我们基于先前的结果，报告了离散可积方程模素数$p$约化理论中的最新成果。本文主要关注Painlevé方程的$q$-离散类比。我们首先简要回顾‘几乎良好约化’这一概念，该性质是大多数在有限域$F_p$上定义的、由$p$-adic数域$Q_p$上方程约化而来的可积系统所具备的。这一概念已在我们的先前论文中引入，并且离散和$q$-离散Painlevé II方程已被证明具有‘几乎良好约化’，这可解释为奇点约束检验的算术类比。在本文中，我们证明该性质对若干其他$q$-离散Painlevé方程也成立。我们还回顾了丢番图可积性理论，并讨论了其与代数熵可积性检测的关系。
英文摘要
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.
发表机构
- Faculty of Engineering Science, Kansai University（关西大学工学部）
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