The paper proves upper bounds on the number of matrices with entries from a finite subset of a finite-rank multiplicative group that have a given rank, determinant, or characteristic polynomial.
Number of characteristic polynomials of matrices with bounded height
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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).
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Counting matrices over finite rank multiplicative groups
The paper proves upper bounds on the number of matrices with entries from a finite subset of a finite-rank multiplicative group that have a given rank, determinant, or characteristic polynomial.