{
  "abstract": "Let $p$ be a large prime and $χ$ a non-trivial Dirichlet character modulo $p$. We study the character sum \\[ \\sum_{a \\sim A} \\sum_{b \\sim B} \\sum_{c \\sim C} α(a,b) β(c)χ(f(a) + bc), \\] where $z \\sim Z$ means $Z\\le z < 2Z$, $f \\in \\mathbb{Z}[X]$ is of small degree $k$, and $\\boldsymbolα=(α(a,b))_{a\\sim A,b\\sim B}$ and $\\boldsymbolβ=(β(c))_{c\\sim C}$ are two complex coefficients. We prove non-trivial upper bounds for this sum in either of the two cases: (1) $\\boldsymbolβ\\equiv1$, $k=2,3,4,5$ and $A,B,C>p^{\\frac{1}{8}+\\varepsilon}$, (2) $\\boldsymbolβ$ general, $k=2,3$ and $A,B,C>p^{\\frac{1}{6}+\\varepsilon}$, where $\\varepsilon>0$ is fixed. This work was originally motivated by an intermediate result of Ganguly and Rajan (2023) on counting $2\\times2$ matrices over $\\mathbb{F}_p$ with irreducible characteristic polynomials, where the entries are in short segments. The new bounds here allow us to count such matrices in much shorter segments.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01524",
  "author_total": 3,
  "authors": [
    {
      "name": "Étienne Fouvry"
    },
    {
      "name": "Igor E. Shparlinski"
    },
    {
      "name": "Ping Xi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8pkyc6x8rmycf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01524",
  "fulltext_url": null,
  "id": "8pkyc6x8rmycf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01524",
  "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",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7215695421",
  "pdf_url": "https://arxiv.org/pdf/2610.01524",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8pkyc6x8rmycf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.01524",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01524"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Tridimensional character sums with polynomial arguments and applications",
  "title_text": "Tridimensional character sums with polynomial arguments and applications",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/8pkyc6x8rmycf",
  "venue": "arXiv",
  "versions": []
}