{
  "abstract": "Using differential Galois theory---specifically, Magid's theory of the complete Picard-Vessiot closure---we show that a system of first-order differential equations in $n$ variables admits a Liouvillian first integral if and only if it admits a first integral in a Liouvillian Picard-Vessiot extension, if and only if it admits a first integral in its Picard-Vessiot ring. Consequently, a system with a Liouvillian first integral admits one in a differential field obtained from the field of rational functions by first taking a finite algebraic extension, then adjoining an exponential of an integral, and then an integral. This yields a new proof of Singer's theorem on Liouvillian first integrals of autonomous planar systems (Trans.\\ Amer.\\ Math.\\ Soc.\\ 333, 1992) and of its recent extension to autonomous systems in $n$ variables by Aziz et al. (arXiv:2512.15522). Our results hold over any differential field of characteristic zero with an algebraically closed field of constants; in particular, they apply to non-autonomous systems. Finally, we exhibit a system that admits a first integral in a non-Liouvillian Picard-Vessiot extension but none in its Picard-Vessiot ring and thus establishing that the Liouvillian hypothesis cannot be dropped.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08623",
  "author_total": 3,
  "authors": [
    {
      "name": "Partha Kumbhakar"
    },
    {
      "name": "Chitrarekha Sahu"
    },
    {
      "name": "Varadharaj R. Srinivasan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0m5j0csqh52zy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08623",
  "fulltext_url": null,
  "id": "0m5j0csqh52zy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08623",
  "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.AC"
    ],
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2608|Advanced Differential Equations and Dynamical Systems"
    ]
  ],
  "openalex": "W7220892243",
  "pdf_url": "https://arxiv.org/pdf/2610.08623",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0m5j0csqh52zy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.08623",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08623"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Liouvillian first integrals of differential equations: A Galoisian approach",
  "title_text": "Liouvillian first integrals of differential equations: A Galoisian approach",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/0m5j0csqh52zy",
  "venue": "arXiv",
  "versions": []
}