{
  "abstract": "Let $K$ be a fixed number field of degree $d$. We prove two determinant-method results for integral points over $\\mathcal O_K$ in boxes whose coordinates may have different size bounds: a Bombieri-Pila type estimate for affine plane curves that retains the two separate coordinate bounds, and a Salberger-Browning covering theorem for affine surfaces. If $F\\in\\mathcal O_K[X_1,X_2]$ is irreducible and $T=\\max B_1^{e_1}B_2^{e_2}$ over the monomials occurring in $F$, then \\[ N_K(F;B_1,B_2)\\ll T^\\varepsilon \\exp\\!\\left(d\\frac{\\log B_1\\log B_2}{\\log T}\\right). \\] For an absolutely irreducible surface in three variables we obtain a covering by $O(T^\\varepsilon V_{3,K})$ bounded-degree auxiliary hypersurfaces, outside $O(T^\\varepsilon V_{3,K}^2)$ exceptional integral points, where \\[ V_{3,K}=\\exp\\!\\left(d \\left(\\frac{\\log B_1\\log B_2\\log B_3}{\\log T}\\right)^{1/2}\\right). \\] These are the number-field analogues of determinant-method estimates for unequal coordinate ranges developed in the rational setting by Browning and Heath-Brown, Salberger, Browning, and Chow-Dietmann. We then apply them to Galois resolvents of monic degree-$n$ polynomials over $K$. Independently, a large-sieve argument, in the tradition of Gallagher's probabilistic Galois theory, gives $O_{K,n}(N^{dn-d/2}\\log N)$ exceptional separable polynomials, while effective Hilbert irreducibility over global fields gives subgroup-sensitive bounds with exponent $d(n-1+1/[S_n:G])$, up to logarithms. We record the interaction with the discriminant-square equation for even groups, congruence restrictions, and rectangular coefficient boxes. All determinant estimates used in the applications are proved in the paper; no conjectural global-field determinant statement is assumed.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.35030",
  "author_total": 1,
  "authors": [
    {
      "name": "Ilaria Viglino"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b8kvnfa1kpwj3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35030",
  "fulltext_url": null,
  "id": "b8kvnfa1kpwj3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.35030",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7224001846",
  "pdf_url": "https://arxiv.org/pdf/2610.35030",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b8kvnfa1kpwj3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.35030",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35030"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Galois groups of polynomials over a fixed number field: determinant methods in general boxes, resolvents, and quantitative counting",
  "title_text": "Galois groups of polynomials over a fixed number field: determinant methods in general boxes, resolvents, and quantitative counting",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/b8kvnfa1kpwj3",
  "venue": "arXiv",
  "versions": []
}