{
  "abstract": "Let $V$ be a DVR with quotient field $K$ and perfect residue field, $v$ be the valuation on $K$ associated with $V$, $\\widehat K$ be the completion of $K$, and $\\mathbb{K}$ be the completion of an algebraic closure $\\overline{\\widehat{K}}$ of $\\widehat K$. We show that a DVR of the rational function field $K(X)$ which is a residually algebraic extension of $V$ is necessarily of the form $V_α=\\{ϕ\\in K(X)\\mid v(ϕ(α))\\geq0\\}$, for an element $α$ of $\\mathbb{K}$ transcendental over $K$, and that $α$ is algebraic over $\\widehat K$ if and only if the residue field extension is finite. Not every such $V_α$ is a DVR, however, and we characterize the $α\\in\\mathbb{K}$ for which $V_α$ is a DVR: they are the elements which can be approximated by algebraic elements in $\\overline{\\widehat{K}}$ with bounded ramification indexes. Combining the two results, we obtain a complete description of the extensions of $V$ to $K(X)$ which are DVRs and residually algebraic over $V$, together with a criterion for each of the two cases to occur. The proofs rest on a bound for the ramification index in a compositum, valid with no tameness assumption and under a separability hypothesis on one residue field extension only; we show that the inequality cannot be improved to a divisibility and that this hypothesis cannot be dropped. We also show that the hypothesis of discreteness cannot be omitted. Furthermore, we show that the set of $α\\in\\mathbb K$ for which $V_α$ is a DVR is a subfield of $\\mathbb K$, which sits properly between $\\overline{\\widehat{K}}$ and $\\mathbb K$, and corresponds to those elements $α$ for which the value group of $\\widehat K(α)$ is discrete.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30483",
  "author_total": 2,
  "authors": [
    {
      "name": "Gyu Whan Chang"
    },
    {
      "name": "Giulio Peruginelli"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dn4k1t5t6n104.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30483",
  "fulltext_url": null,
  "id": "dn4k1t5t6n104",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30483",
  "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",
      "math.AC"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7204922714",
  "pdf_url": "https://arxiv.org/pdf/2608.30483",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dn4k1t5t6n104.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.30483",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30483"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Approximating DVRs by elements of bounded ramification",
  "title_text": "Approximating DVRs by elements of bounded ramification",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/dn4k1t5t6n104",
  "venue": "arXiv",
  "versions": []
}