{
  "abstract": "This paper is motivated by the recent work of Adamczewski and Faverjon [5], who established the algebraic independence of values of Mahler functions satisfying linear functional equations. While their work is a landmark result, some proofs are long and complex. The primary aim of this paper is to simplify those proofs and provide new applications. In their proofs, a vanishing theorem for polynomials, rather than power series, is sufficient. We give a proof of the vanishing theorem without using Szemerédi's theorem or Corvaja-Zannier's theorem. In the case of a single transformation, we present a refined vanishing theorem, which allows us to simplify the proof of algebraic independence. We also provide an elementary proof of a vanishing theorem for $q_i$-Mahler functions with various $q_i$. As an application, we prove the algebraic independence of certain continued fractions, extending previous results by Hančl, Adamczewski, and Bundschuh and Väänänen in which only linear independence and transcendence had been established.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.23435",
  "author_total": 3,
  "authors": [
    {
      "name": "Kumiko Nishioka"
    },
    {
      "name": "Taka-aki Tanaka"
    },
    {
      "name": "Yusuke Tanuma"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/85sv7g8bpbbd3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.23435",
  "fulltext_url": null,
  "id": "85sv7g8bpbbd3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.23435",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7222936116",
  "pdf_url": "https://arxiv.org/pdf/2610.23435",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/85sv7g8bpbbd3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.23435",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.23435"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Mahler Method and Vanishing Theorem",
  "title_text": "Mahler Method and Vanishing Theorem",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/85sv7g8bpbbd3",
  "venue": "arXiv",
  "versions": []
}