{
  "abstract": "For $f(X)=X^r(X^s+a)\\in\\mathbb F_{p^m}[X]$, with $m,r,s\\geqslant1$ and $a\\in\\mathbb F_{p^m}^\\times$, put $g=\\gcd(s,p^m-1)$. We prove that, for each fixed $g$, only finitely many prime characteristics $p$ admit such permutation binomials, even as $m,r,s,a$ vary. If $6\\nmid g$, every such $p$ divides Wendt's determinant $W_g=\\operatorname{Res}_X(X^g-1,(X+1)^g-1)$, which is nonzero. When $6\\mid g$, this determinant vanishes; inverse power sums and iterated resultants yield an explicit nonzero integer divisible by every such $p>2g+1$. We determine the exact admissible characteristics for $1\\leqslant g\\leqslant10$, obtain a further divisibility condition over prime fields, and show that in prime fields $g=11$ occurs precisely when $p=23$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18584",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhichao Tang"
    },
    {
      "name": "Xiang Fan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/as4y6rhxj3ber.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18584",
  "fulltext_url": null,
  "id": "as4y6rhxj3ber",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18584",
  "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",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7221801937",
  "pdf_url": "https://arxiv.org/pdf/2610.18584",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/as4y6rhxj3ber.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:20Z",
      "native_id": "2610.18584",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18584"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Wendt's determinant and finiteness of prime characteristics for permutation binomials",
  "title_text": "Wendt's determinant and finiteness of prime characteristics for permutation binomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:20Z",
  "url": "https://research.draconitic.com/paper/as4y6rhxj3ber",
  "venue": "arXiv",
  "versions": []
}