{
  "abstract": "Let $f\\in\\mathbb Z[x]$ be a polynomial of degree $n$ with integer coefficients of absolute value less than $2^b$. Using Pan's approximate factorization theorem, we give a deterministic algorithm that finds all distinct integer roots of $f$ in $\\widetilde O(nb)$ bit operations. This matches the bit size of the input up to logarithmic factors and does not require $f$ to be square-free. Since we were unable to verify parts of the running time analysis of Pan's theorem, we also give a self-contained deterministic algorithm with running time $\\widetilde O(nb+\\min(n,b)^3b^2)$, which is $\\widetilde O(nb)$ whenever $b\\leq n^{1/4}$. This answers a question of Harvey and Hittmeir (2022), who asked whether integer roots can be found deterministically in quasilinear time, conditionally on Pan's theorem, and unconditionally for $b\\leq n^{1/4}$. The fastest previous deterministic algorithm, due to Nir, takes $\\widetilde O(n^{3/2}b)$ bit operations and works only for square-free polynomials. Instead of $p$-adic lifting, we combine approximate complex root finding with exact polynomial evaluation. We first approximate the complex roots of the polynomial formed by the top $\\min(n,b)+1$ coefficients of $f$. Every integer root $r$ of $f$ with $|r|\\geq 2$ lies within distance $2$ of some complex root of this truncated polynomial. Rounding the real parts of the approximate roots therefore gives a list of $O(\\min(n,b))$ candidate integers that is guaranteed to contain all integer roots of $f$. Although a single candidate can have up to $b$ bits, a bound via the Mahler measure shows that the bit lengths of all candidates sum to $O(b)$. This allows us to test all candidates by exact multipoint evaluation in $\\widetilde O(nb)$ bit operations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.04730",
  "author_total": 2,
  "authors": [
    {
      "name": "Yansong Feng"
    },
    {
      "name": "Itamar Nir"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2kwyjsbknx4y2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.04730",
  "fulltext_url": null,
  "id": "2kwyjsbknx4y2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.04730",
  "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",
      "cs.SC"
    ],
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.04730",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2kwyjsbknx4y2.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.04730",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.04730"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Finding Integer Roots of Polynomials in Nearly Optimal Time",
  "title_text": "Finding Integer Roots of Polynomials in Nearly Optimal Time",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/2kwyjsbknx4y2",
  "venue": "arXiv",
  "versions": []
}