Number of characteristic polynomials of matrices with bounded height
László Mérai, Igor E. Shparlinski
TL;DR
The paper studies the number of distinct irreducible characteristic polynomials (and hence distinct eigenvalues) of $n\times n$ integer matrices with entries bounded by $H$, refining previous bounds by Abrams–Landau–Pommersheim–Srivastava. It constructs explicit matrix families via lower Hessenberg forms and a corresponding family of polynomials, enabling a lifting argument from finite fields to integers. For odd $n\ge 11$ it establishes an asymptotic lower bound ${\#\mathcal I}_n^+(H) \ge (1/n) H^{(n-1)^2/4} (1+o(1))$ as $H\to\infty$, and provides a fully explicit bound valid for all odd $n\ge 5$ and $H\ge 3$, ${\#\mathcal I}_n^+(H) \ge (1/(4n)) H^{(n-1)/2}$ up to lower-order terms. The work also leverages explicit finite-field irreducibility bounds (Pollack) and highlights that for very large $H$, almost all such polynomials are irreducible, yielding the maximal growth rate ${H^{(n-1)^2/4}}$ in that regime. Overall, the paper strengthens the connection between bounded-height integer matrices and irreducible polynomials, yielding sharper exponents and explicit estimates that improve prior results.
Abstract
We consider the set $\mathcal{M}_n(\mathbb{Z}; H)$ of $n\times n$-matrices with integer elements of size at most $H$ and obtain upper and lower bounds on the number of distinct irreducible characteristic polynomials which correspond to these matrices and thus on the number of distinct eigenvalues of these matrices. In particular, we improve some results of A.~Abrams, Z.~Landau, J.~Pommersheim and N.~Srivastava (2022).