{
  "abstract": "We study monic irreducible polynomials over $\\mathbb{F}_q[T]$ whose non-leading digits, with respect to an arbitrary polynomial base $b(T)$, avoid a prescribed set of forbidden digits. Identifying the digit set with the ring $D=\\mathbb{F}_q[T]/(b)$, we obtain an asymptotic formula for the number of such irreducible polynomials under conditions given in terms of Fourier parameters of the allowed digit set. The main term contains a singular series measuring the relative density of units among the allowed digits, while the error term is controlled by both pointwise and averaged Fourier estimates. As a consequence, we obtain a general criterion depending only on the cardinality of the forbidden set, as well as stronger results for structured restrictions, including examples in which the forbidden set contains a positive proportion of all digits. In particular, we treat additive cosets, restrictions compatible with the Chinese remainder decomposition of $D$, and coefficient-wise restrictions. The proof adapts the function field circle method for restricted coefficients to arbitrary polynomial bases.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25367",
  "author_total": 2,
  "authors": [
    {
      "name": "Juan Arévalo"
    },
    {
      "name": "Matilde Lalín"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1p4vef7rrh89c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25367",
  "fulltext_url": null,
  "id": "1p4vef7rrh89c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25367",
  "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.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7214101905",
  "pdf_url": "https://arxiv.org/pdf/2609.25367",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1p4vef7rrh89c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25367",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25367"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Irreducible polynomials with restricted digits in base $b(T)$",
  "title_text": "Irreducible polynomials with restricted digits in base b(T)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/1p4vef7rrh89c",
  "venue": "arXiv",
  "versions": []
}