Galois groups of polynomials over a fixed number field: determinant methods in general boxes, resolvents, and quantitative counting
Abstract
Let K be a fixed number field of degree d. We prove two determinant-method results for integral points over 𝒪K 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∈𝒪K[X1,X2] is irreducible and T=max B1e1B2e2 over the monomials occurring in F, then NK(F;B1,B2)≪ Tε exp (d(log B1log B2)/(log T)). For an absolutely irreducible surface in three variables we obtain a covering by O(Tε V3,K) bounded-degree auxiliary hypersurfaces, outside O(Tε V3,K2) exceptional integral points, where V3,K=exp (d ((log B1log B2log B3)/(log T))1/2). 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 OK,n(Ndn−d/2log N) exceptional separable polynomials, while effective Hilbert irreducibility over global fields gives subgroup-sensitive bounds with exponent d(n−1+1/[Sn: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.
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BibTeX
@misc{viglino2026galois,
title = {{Galois groups of polynomials over a fixed number field: determinant methods in general boxes, resolvents, and quantitative counting}},
author = {Ilaria Viglino},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35030},
eprint = {2610.35030},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35030},
note = {CC0},
}
RIS
TY - UNPB
TI - Galois groups of polynomials over a fixed number field: determinant methods in general boxes, resolvents, and quantitative counting
AU - Ilaria Viglino
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35030
UR - https://arxiv.org/abs/2610.35030
AB - Let K be a fixed number field of degree d. We prove two determinant-method results for integral points over 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∈𝒪_K[X_1,X_2] is irreducible and T=max B_1^e_1B_2^e_2 over the monomials occurring in F, then N_K(F;B_1,B_2)≪ T^ε exp (d(log B_1log B_2)/(log T)). For an absolutely irreducible surface in three variables we obtain a covering by O(T^ε V_3,K) bounded-degree auxiliary hypersurfaces, outside O(T^ε V_3,K^2) exceptional integral points, where V_3,K=exp (d ((log B_1log B_2log B_3)/(log T))^1/2). 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/2log 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.
ER -