Counting matrices with given characteristic polynomial

Abstract: Let $f$ be a monic polynomial of degree $n$ with integer coefficients. We use Fourier analysis to prove a new upper bound on the number of $n\times n$ matrices with characteristic polynomial $f$ and integer entries in $[-T,T]$, as $T\to \infty$. The implied constant in our bound only depends on $n$ and is otherwise uniform in $f$. Our bound improves on recent work of Habegger, Ostafe and Shparlinski for $n\geq 4$.