{
  "abstract": "Let $W_n$ be the Lie algebra of polynomial vector fields in $n$ variables over a field of characteristic zero, and let $s_k$ denote the alternating sum of compositions of $k$ vector fields. The identity $s_{n^2+2n-1}=0$ has been asserted for arbitrary $n$. We give an explicit counterexample in dimension five, verified by an exact integer computation of a $34$-commutator. Together with the known vanishing of $s_{35}$, this shows that $35$ is the least degree of a standard identity on $W_5$. We then prove that $s_{n^2+2n-1}=0$ for every even $n\\geq2$. The proof uses Lie algebra homology with coefficients in the polynomial algebra modulo constants and a lemma on invariant tensors. We describe the computation and provide the source code and verification results.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18149",
  "author_total": 1,
  "authors": [
    {
      "name": "Nurtas Shyntas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/08jvwc942nxw6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18149",
  "fulltext_url": null,
  "id": "08jvwc942nxw6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18149",
  "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.RA"
    ],
    [
      "openalex",
      "2602|Advanced Topics in Algebra"
    ]
  ],
  "openalex": "W7221822521",
  "pdf_url": "https://arxiv.org/pdf/2610.18149",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/08jvwc942nxw6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:20Z",
      "native_id": "2610.18149",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18149"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Standard identities for polynomial vector fields",
  "title_text": "Standard identities for polynomial vector fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:20Z",
  "url": "https://research.draconitic.com/paper/08jvwc942nxw6",
  "venue": "arXiv",
  "versions": []
}