On the sparsity of non-diagonalisable integer matrices and matrices with a given discriminant

Alina Ostafe; Igor E. Shparlinski

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arXiv · 2312.12626

On the sparsity of non-diagonalisable integer matrices and matrices with a given discriminant

Alina Ostafe·Igor E. Shparlinski

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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 bounds on the number of matrices from $\mathcal M_n(\mathbb Z; H)$, for which the characteristic polynomial has a fixed discriminant $d$. When $d=0$, this corresponds to counting matrices with a repeated eigenvalue, and thus is related to counting non-diagonalisable matrices. For $d\ne 0$, this problem seems not to have been studied previously, while for $d=0$, both our approach and the final result improve on those of A. J. Hetzel, J. S. Liew and K. Morrison (2007).

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BibTeX RIS

Alina Ostafe, Igor E. Shparlinski. 2023-12-19. On the sparsity of non-diagonalisable integer matrices and matrices with a given discriminant. https://arxiv.org/abs/2312.12626

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