论文标题
在有限场上的对角线方程
On diagonal equations over finite fields
论文作者
论文摘要
令$ \ mathbb {f} _q $为有限字段，$ q = p^t $元素。在本文中，我们研究表单$ a_1 x_1^{d_1}+\ dots+a_s x_s x_s^{d_s} = b $ over $ \ mathbb {f} _q $的方程式的解决方案数量。 Weil造成的经典whef konwn产生了此类解决方案的束缚。在我们的主要结果中，我们为对角方程的解决方案的数量提供了明确的公式，该方程满足了对指数的某些自然限制。在情况下，$ d_1 = \ dots = d_s $，我们提出了对角方程的解决方案数量的必要条件，相对于Weil的界限而言是最大和最小的。特别是，我们完全表征了最大和最小的费马特类型曲线。
Let $\mathbb{F}_q$ be a finite field with $q=p^t$ elements. In this paper, we study the number of solutions of equations of the form $a_1 x_1^{d_1}+\dots+a_s x_s^{d_s}=b$ over $\mathbb{F}_q$. A classic well-konwn result from Weil yields a bound for such number of solutions. In our main result we give an explicit formula for the number of solutions for diagonal equations satisfying certain natural restrictions on the exponents. In the case $d_1=\dots=d_s$, we present necessary and sufficient conditions for the number of solutions of a diagonal equation being maximal and minimal with respect to Weil's bound. In particular, we completely characterize maximal and minimal Fermat type curves.
