{
  "abstract": "Let $F=(f_1(x_1,\\ldots,x_k),\\ldots, f_m(x_1,\\ldots,x_k))$ be a system of nonconstant polynomials of $k$ variables with integer coefficients and let \\[ {\\gcd}_F(x_1,\\ldots,x_k)= \\gcd(f_1(x_1,\\ldots,x_k),\\ldots, f_m(x_1,\\ldots,x_k)). \\] We obtain an unconditional asymptotic formula for the sum \\[ \\sum_{1\\le x_1,\\ldots,x_k\\le x} h({\\gcd}_F(x_1,\\ldots,x_k)), \\] where $F$ is a system of $m\\ge 2$ polynomials of $k\\ge 2$ variables subject to certain general properties, and $h$ is a bounded strongly multiplicative function. In particular, we deduce an asymptotic formula with error term concerning the density of $k$-tuples of positive integers $(x_1,\\ldots,x_k)$ such that ${\\gcd}_F(x_1,\\ldots,x_k)=1$, given by the Ekedahl-Poonen formula.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28284",
  "author_total": 1,
  "authors": [
    {
      "name": "László Tóth"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bceka8bj5h828.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28284",
  "fulltext_url": null,
  "id": "bceka8bj5h828",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28284",
  "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": "W7214197413",
  "pdf_url": "https://arxiv.org/pdf/2609.28284",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bceka8bj5h828.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.28284",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28284"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Counting $k$-tuples of positive integers such that the values of several polynomials of $k$ variables are relatively prime",
  "title_text": "Counting k-tuples of positive integers such that the values of several polynomials of k variables are relatively prime",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/bceka8bj5h828",
  "venue": "arXiv",
  "versions": []
}