{
  "abstract": "Let $p$ be a prime such that $2$ is a primitive root modulo $p$. We prove that, for every three-element set $L\\subseteq\\mathbb{F}_p$ and every integer $n\\geq1$, there exists a symmetric $n\\times n$ matrix over $\\mathbb{F}_p$ with zero diagonal, off-diagonal entries in $L$, and rank $O_{p,L}(n^{1/3})$. This answers a question of Bukh for all such primes and, in particular, settles the case $L=\\{1,4,6\\}\\subseteq\\mathbb{F}_{19}$ singled out on his problem list. When $0\\notin L$, a standard polynomial argument shows that the exponent $1/3$ is optimal. The proof combines a geometric construction in characteristic two with an incidence matrix factorization over $\\mathbb{F}_p$. The same construction yields integer lifts $L'$ of the prescribed sets $L$ and symmetric $L'$-matrices of rational rank $O_{p,L}(n^{1/3})$. When $0\\notin L'$, this improves Bukh's general quadratic lower bound, giving matrices of order $Ω_{p,L}(r^3)$ and rational rank at most $r$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39908",
  "author_total": 1,
  "authors": [
    {
      "name": "Firdavs Rakhmonov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e9br25y5ercn3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39908",
  "fulltext_url": null,
  "id": "e9br25y5ercn3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39908",
  "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.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39908",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e9br25y5ercn3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39908",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39908"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Low-rank matrices with three prescribed off-diagonal values over $\\mathbb{F}_p$",
  "title_text": "Low-rank matrices with three prescribed off-diagonal values over 𝔽_p",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:38Z",
  "url": "https://research.draconitic.com/paper/e9br25y5ercn3",
  "venue": "arXiv",
  "versions": []
}