Galois groups of random integer matrices

Abstract

We study the number

of integer

n times n n × n $n\times n$

matrices A with entries bounded in absolute value by T such that the Galois group of the characteristic polynomial of A is not the full symmetric group

upper S Subscript n S n $S_n$

. One knows

upper M Subscript n Baseline left parenthesis upper T right parenthesis much greater than upper T Superscript n squared minus n plus 1 Baseline log upper T M n ( T ) ≫ T n 2 − n + 1 log ⁡ T $M_n(T) \gg T^{n^2 - n + 1} \log T$

, which we conjecture is sharp. We first use the large sieve to get

upper M Subscript n Baseline left parenthesis upper T right parenthesis much less than upper T Superscript n squared minus 1 divided by 2 Baseline log upper T M n ( T ) ≪ T n 2 − 1 / 2 log ⁡ T $M_n(T) \ll T^{n^2 - 1/2}\log T$

. 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 .