{
  "abstract": "In the first part of this series the $σ$-ring spanned by the periodic solutions of a generalized linear cellular automaton in a discrete group was shown to be Hopf--Galois, with pro-algebraic Galois group, while a Galois theory for the $σ$-ring spanned by the finite support solutions was left open, with parameterized difference Galois theory suggested as the tool. We apply that tool to automata over $\\mathbb Z$. The Fourier transform turns the automaton into a rank one equation $σ(\\hat x)=\\hatα\\hat x$ over a field on which $δ=z\\,d/dz$ acts, and we compute, and algorithmically decide, its parameterized Galois group: for genuine automata with symbols rational in the time variable it is either $\\mathbb{G}_m(\\mathbb C)$ or $\\mathbb{G}_m$, the dividing line being separability of the symbol, $\\hatα(z,t)=γ(z)ψ(t)$. Finally, the parameterized group in the time direction decides holonomy in the space direction; as a corollary, the Stirling numbers of the first kind are not spatially holonomic.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12280",
  "author_total": 2,
  "authors": [
    {
      "name": "David Blázquez-Sanz"
    },
    {
      "name": "Mario Alejandro Vergara Tapiero"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0ymkb34j3fb7g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12280",
  "fulltext_url": null,
  "id": "0ymkb34j3fb7g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12280",
  "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.AG"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "1703|Cellular Automata and Applications"
    ]
  ],
  "openalex": "W7221081346",
  "pdf_url": "https://arxiv.org/pdf/2610.12280",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0ymkb34j3fb7g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.12280",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12280"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Generalized linear cellular automata in groups and difference Galois theory II",
  "title_text": "Generalized linear cellular automata in groups and difference Galois theory II",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/0ymkb34j3fb7g",
  "venue": "arXiv",
  "versions": []
}