{
  "abstract": "Let $k=\\mathbb{F}_q$, $E=\\mathbb{F}_{q^n}$ and $\\mathrm{Tr}=\\mathrm{Tr}_{E/k}$. For $r\\ge 2$, $a\\in k^{\\times}$ and $x\\in E^{\\times}$, let $\\mathrm{N}(E,r,x,a)$ be the number of $r$-tuples $(x_1,\\cdots,x_r)$ in $(E^{\\times})^r$ satisfying $x_1\\cdots x_r=x$ and $\\mathrm{Tr}(x_1+\\cdots+x_r)=a$. We prove $\\left|\\mathrm{N}(E,r,x,a)-\\left((q^n-1)^{r-1}+(-1)^r\\right)/q\\right|\\le (r^n-1) q^{\\frac{(r-1)n-1}{2}}$. This proves the square-root estimate predicted in Wan's conjecture and generalizes a previous result of Moisio and Wan. For a finite semisimple algebra $B=\\prod\\limits_{i=1}^s M_{d_i}(\\mathbb{F}_{q^{n_i}})$ over $k$ and a regular element $x\\in B^{\\times}$, the same method combined with Zelingher's formula leads to analogous square-root estimates.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.23275",
  "author_total": 3,
  "authors": [
    {
      "name": "Xuejun Guo"
    },
    {
      "name": "Chen Lin"
    },
    {
      "name": "Chenhao Tang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9w0qpap3b0tt6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.23275",
  "fulltext_url": null,
  "id": "9w0qpap3b0tt6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.23275",
  "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"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7171546120",
  "pdf_url": "https://arxiv.org/pdf/2607.23275",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9w0qpap3b0tt6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2607.23275",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.23275"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Matrix Kloosterman sums and product-trace estimates for semisimple algebras",
  "title_text": "Matrix Kloosterman sums and product-trace estimates for semisimple algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/9w0qpap3b0tt6",
  "venue": "arXiv",
  "versions": []
}