Counting matrices with given characteristic polynomial
T. Browning, F. Gundlach, W. Sawin, ArXiv:2610.18226 (2026).
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Browning, Tim;
Gundlach, FabianLibreCat;
Sawin, Will
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 $n\geq 4$.
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arXiv:2610.18226
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Browning T, Gundlach F, Sawin W. Counting matrices with given characteristic polynomial. arXiv:261018226. Published online 2026.
Browning, T., Gundlach, F., & Sawin, W. (2026). Counting matrices with given characteristic polynomial. In arXiv:2610.18226.
@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, Fabian Gundlach, and Will Sawin. “Counting Matrices with given Characteristic Polynomial.” ArXiv:2610.18226, 2026.
T. Browning, F. Gundlach, and W. Sawin, “Counting matrices with given characteristic polynomial,” arXiv:2610.18226. 2026.
Browning, Tim, et al. “Counting Matrices with given Characteristic Polynomial.” ArXiv:2610.18226, 2026.
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