{
  "abstract": "Let $(F|K,w)$ be an immediate valued function field of transcendence degree one over a rank-one perfect valued field $(K,v)$ of characteristic $p>0$. It is henselian rational if $F^h=K(Y)^h$ for some $Y\\in F^h$. Kuhlmann proved henselian rationality over tame fields; we investigate how far his method extends to perfect fields. Relative approximation degrees are a central ingredient in Kuhlmann's approach. We first complete their theory over henselian fields by proving the existence of the relative approximation degree and constant of every polynomial, including for pseudo-convergent sequences of algebraic type. Using the $j$-invariants of associated monomial valuations, we describe these invariants directly through Taylor expansions and extend the henselian degree bound of Kuhlmann and Vlahu. We next study the Artin--Schreier reduction underlying the henselian rationality argument. Over perfect fields, every polynomial is Artin--Schreier equivalent to one whose relative approximation degree lies in ${1,p}$. We construct an explicit rank-one example showing that $p$ cannot always be reduced to one modulo the Artin--Schreier image of $K[X]$. Nevertheless, reduction to degree one becomes possible in this example after passing to equivalence modulo the Artin--Schreier image of $K(X)^h$, and the resulting Artin--Schreier function field is henselian rational. Finally, assume that $K$ equals its absolute ramification field, and let $L=IC(F|K,w)$ be the relative algebraic closure of $K$ in $F^h$. We prove that $F^h$ is henselian rational over $L$, and that henselian rationality descends to $K$ whenever $L|K$ is finite. This finiteness condition holds whenever some separating transcendental element induces an extension of Type II, yielding henselian rationality in this case.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03451",
  "author_total": 2,
  "authors": [
    {
      "name": "Arpan Dutta"
    },
    {
      "name": "Rumi Ghosh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2p16cgz1yq5s9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03451",
  "fulltext_url": null,
  "id": "2p16cgz1yq5s9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03451",
  "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"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7208749074",
  "pdf_url": "https://arxiv.org/pdf/2609.03451",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2p16cgz1yq5s9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2609.03451",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03451"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Relative approximation degrees and the henselian rationality problem over perfect fields",
  "title_text": "Relative approximation degrees and the henselian rationality problem over perfect fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/2p16cgz1yq5s9",
  "venue": "arXiv",
  "versions": []
}