{
  "abstract": "We give a deterministic algorithm for finding all integer roots of a square-free polynomial $f\\in\\mathbb Z[x]$ of degree $n$ with $\\lVert f\\rVert_\\infty<2^b$. The running time is $$ \\tilde{O}(n^{3/2}b), $$ improving the $\\tilde{O}(n^2b)$ bound of Harvey and Hittmeir (Research in Number Theory, 2022). The algorithm follows the classical $p$-adic framework: find roots modulo a prime $p$, lift them modulo a high power of $p$, and verify the lifted candidates. The main new idea is to avoid searching for a prime for which $f\\bmod p$ is square-free. Instead, we find a prime for which the total multiplicity of repeated roots modulo $p$ is small. This requires lifting repeated roots, which we handle using a weighted lifting tree. We also give a faster deterministic candidate-verification algorithm: given $n$ candidate integers smaller in absolute value than $2^b$, we decide which are roots of $f$ in $$ \\tilde{O}(nb+\\min(n^2,nb^2)) $$ bit operations. Together, these ingredients give the first deterministic subquadratic-in-$n$ improvement for integer root finding in the square-free case.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.00668",
  "author_total": 1,
  "authors": [
    {
      "name": "Itamar Nir"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c042982wcnj6e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.00668",
  "fulltext_url": null,
  "id": "c042982wcnj6e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.00668",
  "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",
      "1703|Polynomial and algebraic computation"
    ]
  ],
  "openalex": "W7172416265",
  "pdf_url": "https://arxiv.org/pdf/2608.00668",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c042982wcnj6e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2608.00668",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.00668"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Faster Deterministic Integer Root Finding for Integer Polynomials",
  "title_text": "Faster Deterministic Integer Root Finding for Integer Polynomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/c042982wcnj6e",
  "venue": "arXiv",
  "versions": []
}