{
  "abstract": "Let Q be a real indefinite quadratic form of type (1, 4) with determinant D \\neq 0. We prove that for every c \\in \\mathbb{R}^5 there exists x \\in \\mathbb{Z}^5 such that 0 < Q(x + c) \\le (8|D|)^{1/5}, and that the constant 8 is optimal. This establishes a conjecture of Bambah, Dumir and Hans-Gill and, together with earlier work, completes the determination of the sharp constants Γ_{r,s} for positive inhomogeneous values of nondegenerate indefinite quadratic forms. The proof combines classical reduction and gap theorems with uniform bounds for circular gaps of quadratic residues. A central step identifies the exceptional parameters in certain resonant reduced families: an explicit ladder of integer witnesses resolves their punctured neighbourhoods, leaving only three rational parameter points, subject to stated coefficient conditions. These are treated by retaining the coupling in a preceding, less reduced form. The remaining covering statements are verified by finite certificates in exact rational arithmetic.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41223",
  "author_total": 1,
  "authors": [
    {
      "name": "Vineet Kahlon"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/06nz3bhhevask.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41223",
  "fulltext_url": null,
  "id": "06nz3bhhevask",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41223",
  "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.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41223",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/06nz3bhhevask.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41223",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41223"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Positive Values of Inhomogeneous Quadratic Forms of Type (1, 4): The Constant Γ_{1,4} = 8",
  "title_text": "Positive Values of Inhomogeneous Quadratic Forms of Type (1, 4): The Constant Γ_{1,4} = 8",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/06nz3bhhevask",
  "venue": "arXiv",
  "versions": []
}