论文标题
具有给定的特征多项式和乘法矩阵的整数矩阵
Integer matrices with a given characteristic polynomial and multiplicative dependence of matrices
论文作者
论文摘要
我们考虑$ n \ times n $ - matrices的集合$ \ MATHCAL {M} _n（\ MATHBB Z; H）$，最多最多$ H $的整数元素，并从$ \ m nationcal {m Mathcal {m} _n（\ Mathbb z; h）$ f ins of $ h $的整数元素与$ \ mathcal的矩阵数（\ Mathbb z; z [x] $，相对于$ f $均匀。这补充了A. Eskin，S。Mozes和N. Shah（1996）的渐近公式，其中$ f $必须是固定且不可修复的。 使用此结果，除其他结果外，我们从$ \ Mathcal {M} _n（\ Mathbb Z; H）$中获得了$ s $ TUPLACE的上限和下限，满足了各种多重关系，包括多重依赖性和限制的子集团的子集团的子组$ \ \ \ \ Mathrm {gl} _n（gl} _n（gl} _n（q）这些问题概括了F. Pappalardi，M。Sha，I。E. Shparlinski和C. L. Stewart（2018）的标量$ n = 1 $中研究的问题，这是由于矩阵的非交换性而明显的区别。 在这些问题的动机上，我们还证明了具有固定特征多项式的各种复杂矩阵的各种特性，包括计算该品种的程度。
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.
