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 $\mathcal O_K$ 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.