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PreprintCC BY 4.0arXiv

Counting matrices with given characteristic polynomial

  • Tim Browning
  • Fabian Gundlach
  • Will Sawin
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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× n matrices with characteristic polynomial f and integer entries in [−T,T], as T→ ∞. 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≥ 4.

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Cite

BibTeXRIS
BibTeX
@misc{browning2026counting,
  title = {{Counting matrices with given characteristic polynomial}},
  author = {Tim Browning and Fabian Gundlach and Will Sawin},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.18226},
  eprint = {2610.18226},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.18226},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Counting matrices with given characteristic polynomial
AU  - Tim Browning
AU  - Fabian Gundlach
AU  - Will Sawin
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.18226
UR  - https://arxiv.org/abs/2610.18226
AB  - 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× n matrices with characteristic polynomial f and integer entries in [−T,T], as T→ ∞. 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≥ 4.
ER  -