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Number of characteristic polynomials of matrices with bounded height

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arxiv 2310.09052 v2 pith:4JKHT3AB submitted 2023-10-13 math.NT

László Mérai , Igor E. Shparlinski This is my paper · ORCID
classification math.NT
keywords matricesnumbercharacteristicdistinctpolynomialsabramsboundedbounds
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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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  1. Counting matrices over finite rank multiplicative groups

    math.NT 2025-02 accept novelty 6.0 of 10

    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.

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