{
  "abstract": "We say that a matrix over a finite field $\\mathbb{F}_q$ is positive definite if it is symmetric and each of its leading principal minors is a nonzero square in $\\mathbb{F}_q$. In previous work of the authors [J. Algebra, 2025], the entrywise positivity preservers on $M_n(\\mathbb{F}_q)$ were classified in every case except when $n=2$, $q\\equiv1\\pmod4$, and $q$ is not a square. We settle this remaining case, thereby completing the classification of entrywise positivity preservers over every finite field and in every dimension $n\\ge2$. Our proof is based on a novel idempotent reduction that not only resolves the remaining case but also yields a self-contained proof of the complete classification, while avoiding several technical results used in the earlier arguments. As a further application of the same reduction, we classify the entrywise preservers of strongly nonsingular matrices, i.e., matrices whose leading principal minors are all nonzero. We also prove a more general theorem in odd characteristic: for every prescribed sign pattern of nonzero leading principal minors of matrices of a fixed dimension $n\\ge2$, the entrywise preservers are precisely the positive scalar multiples of field automorphisms. Thus, in odd characteristic, preserving any nonzero leading-principal-minor sign pattern surprisingly forces the preservation of every such sign pattern.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.18978",
  "author_total": 4,
  "authors": [
    {
      "name": "Dominique Guillot"
    },
    {
      "name": "Himanshu Gupta"
    },
    {
      "name": "Prateek Kumar Vishwakarma"
    },
    {
      "name": "Chi Hoi Yip"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6zt8zgpe5e6nx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.18978",
  "fulltext_url": null,
  "id": "6zt8zgpe5e6nx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.18978",
  "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.CO"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7203867582",
  "pdf_url": "https://arxiv.org/pdf/2608.18978",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6zt8zgpe5e6nx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2608.18978",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.18978"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Positivity preservers over finite fields II",
  "title_text": "Positivity preservers over finite fields II",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/6zt8zgpe5e6nx",
  "venue": "arXiv",
  "versions": []
}