{
  "abstract": "We prove a new effective bound for cubic Thue--Mahler equations with power-saving dependence on the regulator. This has some applications. First, we improve Stark's bound $\\log \\max\\{|x|,|y|\\} \\ll_ε|k|^{1+ε}$ for the integer solutions of Mordell's equation $y^2=x^3+k$ ($k$ a non-zero integer) by reducing the exponent $1+ε$ to $1/2+ε$; this is the first power-saving improvement without restrictions on $k$ in more than 50 years. Secondly, for integer squares and cubes of size $\\asymp T$ we improve the known unconditional separation lower bound $(\\log T)^{1-o(1)}$ obtained by Stark in 1973 to $(\\log T)^{2-o(1)}$. Finally, we obtain a power-saving improvement in the conductor aspect of the strongest currently available bounds for Frey's height conjecture (a strengthening of Szpiro's conjecture) in the case of elliptic curves over $\\mathbb{Q}$ with integral $j$-invariant.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.23559",
  "author_total": 1,
  "authors": [
    {
      "name": "Hector Pasten"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e7xxetgktm1q9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.23559",
  "fulltext_url": null,
  "id": "e7xxetgktm1q9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.23559",
  "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",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7204171943",
  "pdf_url": "https://arxiv.org/pdf/2608.23559",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e7xxetgktm1q9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2608.23559",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.23559"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Power-saving bounds for Thue--Mahler and Mordell equations",
  "title_text": "Power-saving bounds for Thue--Mahler and Mordell equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/e7xxetgktm1q9",
  "venue": "arXiv",
  "versions": []
}