Let q be a prime power, let ${\mathbb{F}}_q$ be the finite field with q elements and let $d_1,\dots ,d_k$ be positive integers. In this note we explore the number of solutions $(z_1,\dots ,z_k)\in {\bar{\mathbb{F}}}_q^k$ of the equation$L_1(x_1)+\cdots +L_k(x_k)=b,$ 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 $b\in {\bar{\mathbb{F}}}_q$. 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 ${\mathbb{F}}_{q^n}$.

令q为素数，令${\mathbb{F}}_q$是具有q个元素的有限域，并令$d_{\mathrm{1个}}，\dots ，d_{ķ}$是正整数。在本说明中，我们探讨了许多解决方案$（{ž}_{\mathrm{1个}}，\dots ，{ž}_{ķ}）\in {\stackrel{〜}{\mathbb{F}}}_q^{ķ}$ 等式的${\mathrm{大号}}_{\mathrm{1个}}（X_{\mathrm{1个}}）+\cdots +{\mathrm{大号}}_{ķ}（X_{ķ}）=b，$ 有限制 ${ž}_{\mathrm{一世}}\in {\mathbb{F}}_{q^{d_{\mathrm{一世}}}}$，每个 ${\mathrm{大号}}_{\mathrm{一世}}（X）$ 是形式为非零的多项式 ${\sum }_{Ĵ=0}^{{米}_{\mathrm{一世}}}{\mathrm{一种}}_{\mathrm{一世}Ĵ}{X}^{q^{Ĵ}}\in {\mathbb{F}}_q[X]$ 和 $b\in {\stackrel{〜}{\mathbb{F}}}_q$。我们对元素b进行特征化，上面的方程式有一个解，并且在肯定的情况下，我们确定解的确切数目。作为我们主要结果的应用，我们获得了和集的基数$\sum _{\mathrm{一世}=\mathrm{1个}}^{ķ}{\mathbb{F}}_{q^{d_{\mathrm{一世}}}}：=\{{\alpha }_{\mathrm{1个}}+\cdots +{\alpha }_{ķ}|{\alpha }_{\mathrm{一世}}\in {\mathbb{F}}_{q^{d_{\mathrm{一世}}}}\}。$ 我们的方法还使我们能够解决另一个有趣的问题，涉及元素中元素的存在和数量 ${\mathbb{F}}_{q^{ñ}}$ 在中间有规定的痕迹 ${\mathbb{F}}_q$-扩展 ${\mathbb{F}}_{q^{ñ}}$。