{
  "abstract": "Let $A, B \\in \\mathbb{Z}[x_1, \\dots, x_n]$ be multivariate polynomials with integer coefficients and let $G = \\gcd(A, B)$. We present an algorithm for computing $G$ whose expected bit complexity is asymptotically linear in all fundamental parameters: the number of variables $n$, the term count $T = \\max\\{\\|A\\|_0, \\|B\\|_0, \\|G\\|_0\\}$, the total degree $D$, and the logarithmic coefficient sizes $\\log\\Hi$ and $\\log\\Ho$, where $\\Hi$ bounds the coefficients of the inputs and $\\Ho$ bounds those of the GCD. The bit complexity is characterized by the clean bound \\[ \\widetilde{O}\\bigl( n \\cdot T \\cdot D \\cdot \\log\\Hi \\cdot \\log\\Ho \\bigr). \\] To our knowledge, this is the first sparse GCD algorithm over the integers that achieves linear complexity in all these parameters simultaneously. The integer algorithm is built upon a new field GCD algorithm. For $A, B \\in \\K[x_1, \\dots, x_n]$ over a field $\\K$ with $\\operatorname{char}(\\K) = 0$ or $\\operatorname{char}(\\K) > °G$, we give the first algorithm that computes $G = \\gcd(A,B)$ with expected \\[ \\widetilde{O}\\bigl( n \\cdot T \\cdot D \\bigr) \\] field operations, which is both input- and output-sensitive. The key technical contribution behind both algorithms is a derivative-aided separated Hensel lifting technique introduced in this paper. By introducing an auxiliary variable and leveraging derivative information, our scheme extracts all partial exponents via a single $z^2$-lift per variable, achieving constant sequential depth $O(1)$. This stands in sharp contrast to classical Hensel lifting, which requires $O(D)$ sequential lifting steps and suffers from representation densification in the sparse setting. The field algorithm is then extended to the integer case through modular reduction and rational reconstruction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08074",
  "author_total": 2,
  "authors": [
    {
      "name": "Qiao-Long Huang"
    },
    {
      "name": "Xiao-Shan Gao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4zh0hx3qmsc1n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08074",
  "fulltext_url": null,
  "id": "4zh0hx3qmsc1n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08074",
  "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",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.AC"
    ],
    [
      "openalex",
      "1703|Polynomial and algebraic computation"
    ]
  ],
  "openalex": "W7212079361",
  "pdf_url": "https://arxiv.org/pdf/2609.08074",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4zh0hx3qmsc1n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.08074",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08074"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Sparse Polynomial GCD Algorithms Asymptotically Linear in All Fundamental Parameters",
  "title_text": "Sparse Polynomial GCD Algorithms Asymptotically Linear in All Fundamental Parameters",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/4zh0hx3qmsc1n",
  "venue": "arXiv",
  "versions": []
}