Matrix analogue of the van der Waerden conjecture

Prove the bound \mathsf{M}_n(T)\ll_n T^{n^2-n+1}\log T for the number \mathsf{M}_n(T) of integer n\times n matrices of height at most T whose characteristic polynomial has Galois group different from S_n, including reducible characteristic polynomials.

Background

The quantity \mathsf{M}_n(T) counts integer matrices with entries bounded by T whose characteristic polynomial fails to have the full symmetric Galois group S_n. This includes matrices with reducible characteristic polynomials as well as irreducible characteristic polynomials having a proper Galois group.

The paper proves a bound of the form T^{n^2-1+\rho_n+\varepsilon} for prime n\geq5, improving the previously known large-sieve estimate T^{n^2-1/2}\log T. The resulting exponent is still generally weaker than the conjectured exponent n^2-n+1, so the cited conjecture remains unresolved by the results presented.