Counting Integral Matrices with Given Cubic Characteristic Polynomial

We show that the number of $3\times 3$ matrices with given characteristic polynomial and entries in $\mathbb{Z}\cap [-H,H]$ is at most $c H^{3}(\log H)^{6}$ for all $H\ge 2$, where $c>0$ is an absolute constant.

Authors

Publication Details

Journal
arXiv (Cornell University)
Published
2026-10-26
DOI
https://doi.org/10.48550/arxiv.2610.34973
Primary Topic
Analytic Number Theory Research
Type
preprint
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preprint

Counting Integral Matrices with Given Cubic Characteristic Polynomial

Philipp Habegger
arXiv (Cornell University)
Analytic Number Theory Research
preprint

Counting Integral Matrices with Given Cubic Characteristic Polynomial

Philipp Habegger
preprint en

Abstract

We show that the number of $3\times 3$ matrices with given characteristic polynomial and entries in $\mathbb{Z}\cap [-H,H]$ is at most $c H^{3}(\log H)^{6}$ for all $H\ge 2$, where $c>0$ is an absolute constant.

arXiv (Cornell University)
Analytic Number Theory Research
DOIOpen Access PDFOpenAlex
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Counting Integral Matrices with Given Cubic Characteristic Polynomial — Philipp Habegger · arXiv (Cornell University) (2026) | TGRS Research Map | TGRS