{
  "abstract": "On finite fields of characteristic $p$, PN (perfect nonlinear) functions for odd $p$ and APN (almost perfect nonlinear) functions for even $p$ are well-known classes of highly nonlinear functions. GAPN (generalized almost perfect nonlinear) functions were introduced as a generalization of APN functions for even $p$ to all $p$. One of the main targets of studies on such highly nonlinear functions is their classification. While $p$-exceptional monomial PN and $2$-exceptional monomial APN functions have been classified, the corresponding problem for GAPN functions remains open. Here, a polynomial over $\\mathbb{F}_p$ is called a $p$-exceptional PN (resp. APN, resp. GAPN) function if it is a PN (resp. APN, resp. GAPN) function on $\\mathbb{F}_{p^n}$ for infinitely many positive integers $n$. In this paper, we give a geometric characterization of $p$-exceptional monomial GAPN functions. To this end, for a prime power $q$, we establish a geometric characterization of non-zero polynomials in $\\mathbb{F}_{q}[x,y]$ having no absolutely irreducible factor defined over $\\mathbb{F}_{q}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.08969",
  "author_total": 2,
  "authors": [
    {
      "name": "Masamichi Kuroda"
    },
    {
      "name": "Kentaro Mitsui"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9290a6txmahnf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.08969",
  "fulltext_url": null,
  "id": "9290a6txmahnf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.08969",
  "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.AG"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7202186951",
  "pdf_url": "https://arxiv.org/pdf/2608.08969",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9290a6txmahnf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2608.08969",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.08969"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Geometric characterization of $p$-exceptional monomial GAPN functions",
  "title_text": "Geometric characterization of p-exceptional monomial GAPN functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/9290a6txmahnf",
  "venue": "arXiv",
  "versions": []
}