{
  "abstract": "Let A = F_2[T]. The orthogonal multiplicity m(f) of a monic f in A of degree d >= 1 is the number of g in A with deg g < d, gcd(g,f) = 1, and all partial quotients of g/f of degree one. Blackburn (1998) asked how the proportion p(d) of monic polynomials of degree d with m(f) > 0 behaves as d tends to infinity. We attach to each prime divisor P of f a residue psi_P(f) in F_2 of the differential dT/(T(T+1)f), and prove two unconditional facts about this family: a reciprocity relation, namely that the psi_P(f) sum to zero whenever f is prime to T(T+1), and an exact local distribution, namely that psi_P vanishes for a proportion exactly (|P|-2)/(2(|P|-1)) of the admissible completions when P exactly divides f, and for exactly half of them when P^e exactly divides f with e >= 2. Granting that these local conditions pool independently, we obtain p(d) ~ L d^(-1/2) with L = (9/(4 sqrt(pi))) times the product over the monic irreducibles P of A of degree at least two of (1 + 1/(2|P|)) (1 - 1/|P|)^(1/2), equal to 1.2116120743... The same Euler product evaluated at a second parameter returns the exact identity that the m(f) of a given degree sum to 2^d, which fixes its normalisation. We give the exact value of 2^d p(d) for 1 <= d <= 33; the predicted values agree with these to within 5 10^(-5) for 27 <= d <= 33.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02352",
  "author_total": 1,
  "authors": [
    {
      "name": "David Niedbala Giraudin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1dw1rbpad9jn4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02352",
  "fulltext_url": null,
  "id": "1dw1rbpad9jn4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02352",
  "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": "W7220358051",
  "pdf_url": "https://arxiv.org/pdf/2610.02352",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1dw1rbpad9jn4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.02352",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02352"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The proportion of polynomials over F_2 with positive orthogonal multiplicity",
  "title_text": "The proportion of polynomials over F_2 with positive orthogonal multiplicity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/1dw1rbpad9jn4",
  "venue": "arXiv",
  "versions": []
}