Galois groups of random integer matrices

TL;DR

This paper investigates the frequency of integer matrices with bounded entries whose characteristic polynomial's Galois group is not the full symmetric group, providing bounds and conjectures on their count.

Contribution

It introduces new bounds on the number of such matrices and applies advanced sieve and Fourier analysis techniques to improve these bounds for specific classes.

Findings

Established an upper bound on the count of matrices with non-full symmetric Galois groups.

Conjectured the sharpness of the known lower bound for the count.

Applied Fourier analysis and the geometric sieve to refine bounds for certain matrix classes.

Abstract

We study the number $M_n(T)$ be the number of integer $n\times n$ matrices $A$ with entries bounded in absolute value by $T$ such that the Galois group of characteristic polynomial of $A$ is not the full symmetric group $S_n$. One knows $M_n(T) \gg T^{n^2 - n + 1} \log T$, which we conjecture is sharp. We first use the large sieve to get $M_n(T) \ll T^{n^2 - 1/2}\log T$. Using Fourier analysis and the geometric sieve, as in Bhargava's proof of van der Waerden's conjecture, we improve this bound for some classes of $A$.