{"type":"preprint","external_id":{"arxiv":["2610.18226"]},"date_created":"2026-10-14T08:41:29Z","abstract":[{"lang":"eng","text":"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 $n\\geq 4$."}],"publication":"arXiv:2610.18226","citation":{"apa":"Browning, T., Gundlach, F., &#38; Sawin, W. (2026). Counting matrices with given characteristic polynomial. In <i>arXiv:2610.18226</i>.","ieee":"T. Browning, F. Gundlach, and W. Sawin, “Counting matrices with given characteristic polynomial,” <i>arXiv:2610.18226</i>. 2026.","short":"T. Browning, F. Gundlach, W. Sawin, ArXiv:2610.18226 (2026).","chicago":"Browning, Tim, Fabian Gundlach, and Will Sawin. “Counting Matrices with given Characteristic Polynomial.” <i>ArXiv:2610.18226</i>, 2026.","mla":"Browning, Tim, et al. “Counting Matrices with given Characteristic Polynomial.” <i>ArXiv:2610.18226</i>, 2026.","ama":"Browning T, Gundlach F, Sawin W. Counting matrices with given characteristic polynomial. <i>arXiv:261018226</i>. Published online 2026.","bibtex":"@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} }"},"user_id":"100450","language":[{"iso":"eng"}],"_id":"67523","date_updated":"2026-10-14T08:42:52Z","year":"2026","title":"Counting matrices with given characteristic polynomial","status":"public","author":[{"full_name":"Browning, Tim","first_name":"Tim","last_name":"Browning"},{"id":"100450","first_name":"Fabian","last_name":"Gundlach","full_name":"Gundlach, Fabian"},{"full_name":"Sawin, Will","last_name":"Sawin","first_name":"Will"}]}