Integer matrices with a given characteristic polynomial and multiplicative dependence of matrices
| Autor: | Habegger, Philipp, Ostafe, Alina, Shparlinski, Igor E. |
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| Rok vydání: | 2022 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 a new upper bound on the number of matrices from $\mathcal{M}_n(\mathbb Z; H)$ with a given characteristic polynomial $f \in \mathbb Z[X]$, which is uniform with respect to $f$. This complements the asymptotic formula of A. Eskin, S. Mozes and N. Shah (1996) in which $f$ has to be fixed and irreducible. Using this result, among others, we obtain upper and lower bounds on the number of $s$-tuples of matrices from $\mathcal{M}_n(\mathbb Z; H)$, satisfying various multiplicative relations, including multiplicative dependence and bounded generation of a subgroup of $\mathrm{GL}_n(\mathbb Q)$. These problems generalise those studied in the scalar case $n=1$ by F. Pappalardi, M. Sha, I. E. Shparlinski and C. L. Stewart (2018) with an obvious distinction due to the non-commutativity of matrices. Motivated by these problems, we also prove various properties of the variety of complex matrices with fixed characteristic polynomial, including computing the degree of this variety. Comment: In this new version, now jointly with Philipp Habegger, we use a very different approach to counting integer matrices with a given characteristic polynomial, which leads to a much stronger estimate. In turn, this also improves several other results |
| Databáze: | arXiv |
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