Galois groups of polynomials over a fixed number field: determinant methods in general boxes, resolvents, and quantitative counting

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 \[ N_K(F;B_1,B_2) T^ε exp\!(dlog B_1log B_2/log T). \] For an absolutely irreducible surface in three variables we obtain a covering by O(T^ε V3,K) bounded-degree auxiliary hypersurfaces, outside O(T^ε V3,K²) exceptional integral points, where \[ V3,K=exp\!(d (log B_1log B_2log B_3/log T)1/2). \] 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 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.