{
  "abstract": "The orbit problem is the problem of whether, given $x, y\\in \\mathbb{Q}^n$ and an $n\\times n$ matrix $A$, there exists an $i\\in\\mathbb{N}$ such that $A^ix = y$. In 1980, Kannan and Lipton proved that the orbit problem is decidable. We show that a generalization of the orbit problem, where the field is $\\mathbb{Q}(X)$ for $X$ a countable set of transcendentals, is also decidable. We define the orbit power problem for an arbitrary group $G$ to be the problem of when, given $x, y\\in G$, there exists an $n\\in\\mathbb{Z}$ such that $x^n = y$. We then use the main result to show that the power orbit problem is decidable for an assortment of groups, including the braid groups and $\\operatorname{Aut}(F_2)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29440",
  "author_total": 2,
  "authors": [
    {
      "name": "Joshua Holden"
    },
    {
      "name": "Alexa Renner"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b5r77xpa69gpg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29440",
  "fulltext_url": null,
  "id": "b5r77xpa69gpg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29440",
  "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.GR"
    ],
    [
      "openalex",
      "1703|semigroups and automata theory"
    ]
  ],
  "openalex": "W7204958237",
  "pdf_url": "https://arxiv.org/pdf/2608.29440",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b5r77xpa69gpg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.29440",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29440"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Decidability of the Orbit Problem Over $\\mathbb{Q}(X)$",
  "title_text": "Decidability of the Orbit Problem Over ℚ(X)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/b5r77xpa69gpg",
  "venue": "arXiv",
  "versions": []
}