Additive Decompositions by Conjugacy Classes in $M_n({\mathbb{F}}_q)$

Abstract

Let $n\ge 2$ 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 $(n,q,\text{Tr}(z))=(2,2,1)$, 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 $(T-\lambda )h(T)$, where $\lambda \in {\mathbb{F}}_q$, $h$ is irreducible of degree $n-1$ and $h(\lambda )\ne 0$. These classes can be chosen so that $\text{Tr}(z)=\text{Tr}(X)+\text{Tr}(Y)$. For such $X$ and $Y$, let $$N_{X,Y}(z)=\#\{(x,y)\in X\times Y:x+y=z\}.$$ We prove the following estimate: $$\left|N_{X,Y}(z)-q^{(n-1{)}^2}\right|\le 42q^{(n-1{)}^2-1}.$$ 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 $q$.

21 pages