On the sparsity of non-diagonalisable integer matrices and matrices with a given discriminant
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908910691024896 |
|---|---|
| author | Ostafe, Alina Shparlinski, Igor E. |
| author_facet | Ostafe, Alina Shparlinski, Igor E. |
| contents | 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_12626 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the sparsity of non-diagonalisable integer matrices and matrices with a given discriminant Ostafe, Alina Shparlinski, Igor E. Number Theory 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). |
| title | On the sparsity of non-diagonalisable integer matrices and matrices with a given discriminant |
| topic | Number Theory |
| url | https://arxiv.org/abs/2312.12626 |