{
  "abstract": "Let $f,g\\in\\mathbb{Z}[t]$ be products of cyclotomic polynomials such that either $g/\\gcd(f,g)$ or $f/\\gcd(f,g)$ (or both) is squarefree. We give explicit formulae for the invariant factors of $f(C_g)$, where $C_g$ is the companion matrix of $g$, expressing them in terms of the indices and multiplicities of the cyclotomic factors of $f$ and $g$. A more compact formula holds for the circulant case $g(t)=t^n-1$, assuming that every cyclotomic factor shared by $f$ and $g$ has multiplicity $1$ in $f$. We discuss applications to several topics, including abelianizations of cyclically presented groups, homology groups of cyclic branched covers of knots, adjacency matrices and shifted graph Laplacians, $K$-theory, and Smith forms of matrix powers. For example, for two families of hyperbolic knots we give the first integral homology in every covering degree; or, when $g$ is squarefree, we determine the least eventual period of the Smith forms of powers of $f(C_g)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.19205",
  "author_total": 2,
  "authors": [
    {
      "name": "Vanni Noferini"
    },
    {
      "name": "Gerald Williams"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2271ay3ct41mp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.19205",
  "fulltext_url": null,
  "id": "2271ay3ct41mp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.19205",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GR"
    ],
    [
      "arxiv",
      "math.KT"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7222053153",
  "pdf_url": "https://arxiv.org/pdf/2610.19205",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2271ay3ct41mp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.19205",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.19205"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Invariant factors from products of cyclotomic polynomials, with applications",
  "title_text": "Invariant factors from products of cyclotomic polynomials, with applications",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/2271ay3ct41mp",
  "venue": "arXiv",
  "versions": []
}