{
  "abstract": "We study the distinctness and algebraic degrees of inverted Kloosterman sums over finite extensions of $\\mathbb{F}_p$. For each $d\\geq 2$, we prove that the $(d-1)$-dimensional sums over $\\mathbb{F}_{p^2}$ are equal precisely when their parameters are Frobenius conjugate, provided that $p\\equiv 1\\pmod d$, $p>2d+1$, and $p$ does not divide the nonzero integer $2(d+1)^d-\\binom{2d}{d}^2$. For two-dimensional sums over $\\mathbb{F}_{p^r}$, the same conclusion holds for $2\\leq r\\leq 5$, $p\\equiv 1\\pmod 3$, and $p>3r+1$. In both cases we determine the algebraic degree of every sum in terms of the minimal polynomial of its parameter. The proof combines a cyclic-carry analysis of a Gauss-sum expansion with the recovery of power traces from a formal logarithm.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25443",
  "author_total": 1,
  "authors": [
    {
      "name": "Danyil Kyselov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/07sxtvwhdjswz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25443",
  "fulltext_url": null,
  "id": "07sxtvwhdjswz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25443",
  "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": "W7223053330",
  "pdf_url": "https://arxiv.org/pdf/2610.25443",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/07sxtvwhdjswz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25443",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25443"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Distinctness and Algebraic Degrees of Inverted Kloosterman Sums",
  "title_text": "Distinctness and Algebraic Degrees of Inverted Kloosterman Sums",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/07sxtvwhdjswz",
  "venue": "arXiv",
  "versions": []
}