Counting matrices with given characteristic polynomial Browning, Tim Gundlach, Fabian Sawin, Will 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 $n\geq 4$. 2026 info:eu-repo/semantics/preprint doc-type:preprint text http://purl.org/coar/resource_type/c_816b https://ris.uni-paderborn.de/record/67523 Browning T, Gundlach F, Sawin W. Counting matrices with given characteristic polynomial. <i>arXiv:261018226</i>. Published online 2026. eng info:eu-repo/semantics/altIdentifier/arxiv/2610.18226 info:eu-repo/semantics/closedAccess