{
  "abstract": "We show that there are integers $N$ and odd $D$ such that no algorithm can decide, from the rational coefficients of a homogeneous polynomial vector field $F$ in dimension $N$ of degree $D$, whether the origin is Lyapunov stable for $\\dot Y=F(Y)$. This proves a conjecture of V. I. Arnold. We provide a Lean formalization of the proof. The smallest dimension $N$ for which we were able to prove undecidability is $N = 5$. An analogous undecidability result holds for global asymptotic stability and global exponential stability for non-homogenous vector fields. For dimension $N=2$ and homogenous vector fields, we establish decidability for all commonly used stability notions, building heavily on existing results.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22058",
  "author_total": 1,
  "authors": [
    {
      "name": "Milan Korda"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/82hf4hcjk4r7v.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22058",
  "fulltext_url": null,
  "id": "82hf4hcjk4r7v",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22058",
  "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.CA"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2608|Advanced Differential Equations and Dynamical Systems"
    ]
  ],
  "openalex": "W7213953160",
  "pdf_url": "https://arxiv.org/pdf/2609.22058",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/82hf4hcjk4r7v.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.22058",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22058"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Lyapunov stability of polynomial vector fields is undecidable",
  "title_text": "Lyapunov stability of polynomial vector fields is undecidable",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/82hf4hcjk4r7v",
  "venue": "arXiv",
  "versions": []
}