{
  "abstract": "We study finite-fibre determinants for rational Bishop operators and their role in cyclicity for irrational parameters. The paper has two main parts. First, for the constant vector $f=1$, a resultant identity and a discrete Fourier factorization reveal a determinant parity mechanism for general modular orbit order: odd denominators give a nonnegative normalized determinant on the positive fundamental cell, while for even denominators the unique real alternating Fourier mode is the only factor capable of producing a sign-changing zero. We give an explicit example at $(r,q)=(9,16)$ and an analytic infinite family $(r,q)=(3,6n-2)$. Grivaux's zero-free determinant is identified as the consecutive-order subfamily $D_{1,q}$, so these zeros are caused specifically by nonconsecutive modular ordering. Second, we prove an explicit cyclicity criterion that does not require global nondegeneracy or monotonicity of the fibre determinant. A quantitative Remez estimate controls the small-determinant set; cutoff inverses are approximated by endpoint-corrected Fejer polynomials; and an explicit continuity modulus transfers the resulting rational approximants to irrational parameters. This produces a fully explicit continued-fraction gap function for $f=1$ and, more generally, for every polynomial $f$ with $f(0)\\ne 0$. The argument also gives the exact degree and leading coefficient of the corresponding polynomial-vector fibre determinants.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.27946",
  "author_total": 1,
  "authors": [
    {
      "name": "Yicen Ma"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bges5tqm8t372.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.27946",
  "fulltext_url": null,
  "id": "bges5tqm8t372",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.27946",
  "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.FA"
    ],
    [
      "openalex",
      "2604|Holomorphic and Operator Theory"
    ]
  ],
  "openalex": "W7204856476",
  "pdf_url": "https://arxiv.org/pdf/2608.27946",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bges5tqm8t372.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2608.27946",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.27946"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Rational Bishop determinants and explicit cyclicity criteria",
  "title_text": "Rational Bishop determinants and explicit cyclicity criteria",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/bges5tqm8t372",
  "venue": "arXiv",
  "versions": []
}