preprint Tim Browning author Fabian Gundlach author 100450 Will Sawin author 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 eng 2610.18226 Browning T, Gundlach F, Sawin W. Counting matrices with given characteristic polynomial. <i>arXiv:261018226</i>. Published online 2026. @article{Browning_Gundlach_Sawin_2026, title={Counting matrices with given characteristic polynomial}, journal={arXiv:2610.18226}, author={Browning, Tim and Gundlach, Fabian and Sawin, Will}, year={2026} } Browning, Tim, et al. “Counting Matrices with given Characteristic Polynomial.” <i>ArXiv:2610.18226</i>, 2026. T. Browning, F. Gundlach, W. Sawin, ArXiv:2610.18226 (2026). Browning, Tim, Fabian Gundlach, and Will Sawin. “Counting Matrices with given Characteristic Polynomial.” <i>ArXiv:2610.18226</i>, 2026. Browning, T., Gundlach, F., &#38; Sawin, W. (2026). Counting matrices with given characteristic polynomial. In <i>arXiv:2610.18226</i>. T. Browning, F. Gundlach, and W. Sawin, “Counting matrices with given characteristic polynomial,” <i>arXiv:2610.18226</i>. 2026. 675232026-10-14T08:41:29Z2026-10-14T08:42:52Z