AI 中文总结
该研究针对有限域上n阶矩阵环，证明除特定例外情况外，存在满足迹匹配的共轭类，给出非标量矩阵分解为这两类元素之和的数量估计，其值与q^{(n−1)^2}相近且误差常数与n、q无关。
AI 中文摘要
设n≥2为正整数，q为素数幂，我们研究有限域F_q上的n阶矩阵环M_n(F_q)中非标量矩阵作为两个给定共轭类中元素之和的加法分解的数量。设z∈M_n(F_q)为非标量矩阵，我们证明，除了(n,q,Tr(z))=(2,2,1)的情况外，存在M_n(F_q)中的共轭类X、Y，使得X的特征多项式为n次不可约多项式，Y的特征多项式为(T−λ)h(T)的形式，其中λ∈F_q，h为n−1次不可约多项式且h(λ)≠0；这些类可选取满足Tr(z)=Tr(X)+Tr(Y)。对于这样的X和Y，令N_{X,Y}(z)为满足x+y=z的(x,y)∈X×Y的数量，我们证明估计式：|N_{X,Y}(z)−q^{(n−1)^2}|≤42q^{(n−1)^2−1}。因此，对于迹匹配的非标量矩阵，此类加法分解的数量大致相同，均为q^{(n−1)^2}，且绝对误差常数与n、q无关。
英文摘要
Let $n \geq 2$ be a positive integer, and $q$ be a prime power. We study the $number$ of additive decompositions of nonscalar matrices in the matrix ring $M_n(\mathbb{F}_q)$ as sums of elements from two prescribed conjugacy classes. Let $z \in M_n(\mathbb{F}_q)$ be nonscalar. We show that, except for the case $(n,q,\textrm{Tr}(z)) = (2,2,1)$, there exist conjugacy classes $X, Y \subset M_n(\mathbb{F}_q)$ such that the characteristic polynomial of $X$ is irreducible of degree $n$ and the characteristic polynomial of $Y$ is of the form $ (T- λ) h(T)$, where $λ\in \mathbb{F}_q$, $h$ is irreducible of degree $n-1$ and $h(λ) \neq 0$. These classes can be chosen so that $\textrm{Tr}(z) = \textrm{Tr}(X) + \textrm{Tr}(Y)$. For such $X$ and $Y$, let $$ N_{X,Y}(z) = \# \{ (x,y) \in X \times Y : x + y = z \}. $$ We prove the following estimate: $$ \left| N_{X,Y}(z) - q^{(n-1)^2} \right| \leq 42 q^{(n-1)^2 -1}. $$ Thus, for nonscalar matrices with matching trace, the number of such additive decompositions is $approximately$ the same, namely $q^{(n-1)^2}$, with an absolute error constant independent of $n$ and $q$.
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