{
  "abstract": "This paper is devoted to studying permutation binomial $F_{r,a}(X)=X^r(X^{q-1}+a)\\in\\mathbb F_{q^e}[X]$ with $a\\in\\mathbb F_{q^e}^*$. We present a complete characterization for $F_{r,a}$ to be a permutation of $\\mathbb{F}_{q^e}$. This yields a complete proof of the conjecture proposed by Masuda--Rubio--Santiago \\cite{masuda2022permutation}. More precisely, we show that \\(F_{r,a}\\) permutes \\(\\mathbb F_{q^e}\\) if and only if $(-a)^{(q^e-1)/(q-1)}\\ne1, \\gcd(r,q-1)=1$, and there exists \\(1\\le h<e\\) with \\(\\gcd(h,e)=1\\) such that $r(q^h-1)\\equiv q-1\\pmod{q^e-1}.$",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28595",
  "author_total": 4,
  "authors": [
    {
      "name": "Junna Ni"
    },
    {
      "name": "Xuan Pang"
    },
    {
      "name": "Jianhua Yu"
    },
    {
      "name": "Pingzhi Yuan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3v04pv0y1pkvp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28595",
  "fulltext_url": null,
  "id": "3v04pv0y1pkvp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28595",
  "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",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7214296675",
  "pdf_url": "https://arxiv.org/pdf/2609.28595",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3v04pv0y1pkvp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.28595",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28595"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Permutation binomials of the form $X^r(X^{q-1}+a)$ over finite fields",
  "title_text": "Permutation binomials of the form X^r(X^q−1+a) over finite fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/3v04pv0y1pkvp",
  "venue": "arXiv",
  "versions": []
}