Galois groups of integer polynomials with prescribed coefficients

Abstract

Fix a degree $n$, a nonzero integer constant term, and some further coefficients of a monic integer polynomial, leaving $s$ coefficients to vary in $[-H,H]$. We first prove that, when only the leading and constant coefficients are fixed, the number of polynomials whose Galois group is not $S_n$ is of order $H^{n-2}$ for every $n\ge4$. We then prove a half-dimensional extension: for $n\ge6$ the corresponding exceptional count is of order $H^{s-1}$ whenever $s>n/2+1$, provided the coefficient slice meets the higher-multiplicity loci with a specified dimension bound. This geometric condition holds for every prescribed value if the free coefficients form an initial or final block, and for a nonempty Zariski-open set of prescribed values for arbitrary coefficient positions. It also holds for every value of one additional prescribed interior coefficient. Thus, in degrees seven and eight, any one interior coefficient can be fixed in addition to the leading and constant coefficients. The proof combines a fixed-norm version of Bhargava's primitive-group sieve with sparse Hermite interpolation. A separate imprimitive estimate, using a minimal intermediate field, gives $O(H^{\beta(n)})$ outside rational partial-product hypersurfaces, where $\beta(n)=\ell+n/\ell-2\le n/2$ for composite $n$ with smallest prime divisor $\ell$, and has no logarithmic loss. The fixed-constant quartic and quintic endpoints require separate local arguments: exceptional Fourier directions for $A_4$ quartics and a quadratic Gauss sum for quintics. We isolate further even-degree endpoint results, explain the distinct cubic obstruction, and conclude with conditional improvements under the upper-bound form of Malle's conjecture.