论文标题
计算有限场上特殊线性方程的解决方案
Counting solutions of special linear equations over finite fields
论文作者
论文摘要
令$ q $为主要功率，让$ \ mathbb f_q $是带有$ q $元素的有限字段，让$ d_1，\ ldots，d_k $为正整数。在此注释中，我们探讨了方程{\ Mathbb f} _Q^k $的解决方案$（z_1，\ ldots，z_k）\ in \ intline {\ mathbb f} _q^k $的forkation \ begin \ begin {equication*} l_1（x_1）（x_1）+\ cdots+\ cdots+cdots+l_k（x_k）= b，\ e d diftions = girtions untions priction f_ {q^{d_i}} $，其中每个$ l_i（x）$是$ \ sum_ {j = 0}^{m_i} a_ {ij} a_ {ij} x^x^{q^j^j} \ in \ mathb f_q [x] $和$ b \ $ y \ in的非零多项式。我们表征了上述方程具有解决方案的元素$ b $，在肯定的情况下，我们确定了解决方案的确切数量。作为我们主要结果的应用，我们获得了集合$ \ sum_ {i = 1}^k \ mathbb f_ {q^{q^{d_i}}：= \ {α_1+\ cdots+α_k\，| \ c \ c \ c.我们解决了另一个有趣的问题，关于$ \ mathbb f_ {q^n} $中的元素的存在和数量，并具有$ \ mathbb f_ {q^n} $的中间$ \ mathbb f_q $ - extensions的规定痕迹。
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 $$\sum_{i=1}^k\mathbb F_{q^{d_i}}:=\{α_1+\cdots+α_k\,|\, α_i\in \mathbb F_{q^{d_i}}\}.$$ 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 $\mathbb F_{q^n}$.
