In this paper we study the set of

-rational solutions of equations defined by polynomials evaluated in power-sum polynomials with coefficients in

. This is done by means of applying a methodology which relies on the study of the geometry of the set of common zeros of symmetric polynomials over the algebraic closure of

. We provide improved estimates and existence results of

-rational solutions to the following equations: deformed diagonal equations, generalized Markoff-Hurwitz-type equations and Carlitz's equations. We extend these techniques to more general variants of diagonal equations over finite fields.

中文翻译：

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估计数量 ${\mathbb{F}}_q$–有限域上对角方程变体的有理解

在本文中，我们研究了

由幂和多项式求出的多项式定义的方程的有理解，系数为

。这是通过应用一种方法来完成的，该方法依赖于对多项式的代数闭包上的对称多项式的公零集的几何形状的研究。

。我们提供了改进的估计和存在结果

下列方程的有理解：变形对角线方程，广义Markoff-Hurwitz型方程和Carlitz方程。我们将这些技术扩展到有限域上对角方程的更一般的变体。