In this paper we obtain explicit estimates and existence results on the number of ${\mathbb{F}}_q$-rational solutions of certain systems defined by families of diagonal equations over finite fields. Our approach relies on the study of the geometric properties of the varieties defined by the systems involved. We apply these results to a generalization of Waring's problem and the distribution of solutions of congruences modulo a prime number.

中文翻译：

---

关于有限域上某些对角方程组的解数

在本文中，我们获得了关于数量的明确估计和存在结果${\mathbb{F}}_q$-由有限域上的对角方程族定义的某些系统的有理解。我们的方法依赖于对所涉及系统定义的品种的几何特性的研究。我们将这些结果应用于 Waring 问题的推广和以素数为模的同余解的分布。