{
  "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 $F_p$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33105",
  "author_total": 1,
  "authors": [
    {
      "name": "Masataka Kanki"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/29hmtseb1wsrs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33105",
  "fulltext_url": null,
  "id": "29hmtseb1wsrs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33105",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "nlin.SI"
    ],
    [
      "openalex",
      "3109|Nonlinear Waves and Solitons"
    ]
  ],
  "openalex": "W7214774420",
  "pdf_url": "https://arxiv.org/pdf/2609.33105",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/29hmtseb1wsrs.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:12Z",
      "native_id": "2609.33105",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33105"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:nlin.SI",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Several results on the almost good reduction property to the $q$-discrete Painlevé equations over the field of $p$-adic numbers",
  "title_text": "Several results on the almost good reduction property to the q-discrete Painlevé equations over the field of p-adic numbers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:12Z",
  "url": "https://research.draconitic.com/paper/29hmtseb1wsrs",
  "venue": "arXiv",
  "versions": []
}