{
  "abstract": "Let $D$ be a division ring, let $n\\geq 2$, and let $f,g:D\\to D$ be additive maps satisfying $f(x)x^{-1}+x^n g(x^{-1})=0$ for every nonzero $x\\in D$. We establish general vanishing criteria, give an explicit classification over finite fields, and determine the additive-polynomial solutions over infinite fields of positive characteristic. If $\\mathbb{F}_q\\subseteq Z(D)$, every additive map $D\\to D$ admits a canonical decomposition into $\\mathbb{F}_q^\\times$-weight components, and the identity pairs precisely the weights $r,s$ satisfying $r+s\\equiv n+1\\pmod{q-1}$. Consequently, for $q=p^m$, the dimension over $\\mathbb{F}_q$ of the solution space is the number of ordered pairs $(i,j)$ with $0\\leq i,j<m$ satisfying $p^i+p^j\\equiv n+1\\pmod{q-1}$. Over an infinite field of characteristic $p>0$, the additive-polynomial solutions are exactly the sums of paired Frobenius terms with $p^i+p^j=n+1$. This classification concerns additive-polynomial maps and does not assert a classification of arbitrary additive maps. Prime-field dilation gives complete vanishing in characteristic zero and, in characteristic $p>0$, whenever $p-1$ does not divide $n-1$. In characteristic two, we prove complete vanishing for $n=2$ on every noncommutative division ring. More generally, when $[D:Z(D)]=\\infty$, every solution vanishes if the center is infinite. If the center is the finite field $\\mathbb{F}_q$, the same conclusion holds for $2\\leq n\\leq q-1$. The remaining cases are stated explicitly.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.09669",
  "author_total": 1,
  "authors": [
    {
      "name": "Mohsen Aliabadi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/55hhks85zr146.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.09669",
  "fulltext_url": null,
  "id": "55hhks85zr146",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.09669",
  "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.RA"
    ],
    [
      "openalex",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7168261504",
  "pdf_url": "https://arxiv.org/pdf/2607.09669",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/55hhks85zr146.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2607.09669",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.09669"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Higher-power inverse functional identities and Frobenius collision obstructions",
  "title_text": "Higher-power inverse functional identities and Frobenius collision obstructions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/55hhks85zr146",
  "venue": "arXiv",
  "versions": []
}