---
_id: '67523'
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 $n\geq 4$.
author:
- first_name: Tim
  full_name: Browning, Tim
  last_name: Browning
- first_name: Fabian
  full_name: Gundlach, Fabian
  id: '100450'
  last_name: Gundlach
- first_name: Will
  full_name: Sawin, Will
  last_name: Sawin
citation:
  ama: Browning T, Gundlach F, Sawin W. Counting matrices with given characteristic
    polynomial. <i>arXiv:261018226</i>. Published online 2026.
  apa: Browning, T., Gundlach, F., &#38; Sawin, W. (2026). Counting matrices with
    given characteristic polynomial. In <i>arXiv:2610.18226</i>.
  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} }'
  chicago: Browning, Tim, Fabian Gundlach, and Will Sawin. “Counting Matrices with
    given Characteristic Polynomial.” <i>ArXiv:2610.18226</i>, 2026.
  ieee: T. Browning, F. Gundlach, and W. 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.
  short: T. Browning, F. Gundlach, W. Sawin, ArXiv:2610.18226 (2026).
date_created: 2026-10-14T08:41:29Z
date_updated: 2026-10-14T08:42:52Z
external_id:
  arxiv:
  - '2610.18226'
language:
- iso: eng
publication: arXiv:2610.18226
status: public
title: Counting matrices with given characteristic polynomial
type: preprint
user_id: '100450'
year: '2026'
...