Counting solutions of special linear equations over finite fields

Abstract: Let $q$ be a prime power, let $\mathbb F_q$ be the finite field with $q$ elements and let $d_1, \ldots, d_k$ be positive integers. In this note we explore the number of solutions $(z_1, \ldots, z_k)\in\overline{\mathbb F}q^k$ of the equation \begin{equation*}L_1(x_1)+\cdots+L_k(x_k)=b,\end{equation*} with the restrictions $z_i\in \mathbb F{q^{d_i}}$, where each $L_i(x)$ is a non zero polynomial of the form $\sum_{j=0}^{m_i}a_{ij}x^{q^j}\in \mathbb F_q[x]$ and $b\in \overline{\mathbb F}q$. We characterize the elements $b$ for which the equation above has a solution and, in affirmative case, we determine the exact number of solutions. As an application of our main result, we obtain the cardinality of the sumset $$\sum{i=1}^k\mathbb F_{q^{d_i}}:={\alpha_1+\cdots+\alpha_k\,|\, \alpha_i\in \mathbb F_{q^{d_i}}}.$$ Our approach also allows us to solve another interesting problem, regarding the existence and number of elements in $\mathbb F_{q^n}$ with prescribed traces over intermediate $\mathbb F_q$-extensions of $\mathbb F_{q^n}$.