{
  "abstract": "Let $K$ be a perfectoid field of characteristic $p$ with countable value group (for example, $K=\\mathbb{C}_{p}^{\\flat}$). We study Zariski-closed subsets \\begin{equation*}Z\\subseteq \\mathbb{B}_{K}^{n,\\mathrm{perfd}}\\end{equation*}of the $n$-dimensional perfectoid closed unit ball $\\mathbb{B}_{K}^{n,\\mathrm{perfd}}$ over $K$. We use an example of Gleason, and its generalization obtained in the authors' previous work, to show that two natural notions of dimension for such Zariski-closed subsets, the Krull dimension $\\dim(Z)$ and Huber's iterated transcendence degree $\\mathrm{dim.tr}(Z)$, do not coincide, and that the gap between them can be arbitrarily large.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01418",
  "author_total": 2,
  "authors": [
    {
      "name": "Dimitri Dine"
    },
    {
      "name": "Jack J Garzella"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fqg1nk15xgknz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01418",
  "fulltext_url": null,
  "id": "fqg1nk15xgknz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01418",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AC"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01418",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fqg1nk15xgknz.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01418",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01418"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Topological transcendence degree and the dimension problem for perfectoid spaces",
  "title_text": "Topological transcendence degree and the dimension problem for perfectoid spaces",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/fqg1nk15xgknz",
  "venue": "arXiv",
  "versions": []
}