In this article we establish the asymptotic behavior of generating functions related to the exponential sum over finite fields of elementary symmetric functions and their perturbations. This asymptotic behavior allows us to calculate the probability generating function of the probability that the elementary symmetric polynomial of degree k and its perturbations returns $\beta \in {\mathbb{F}}_q$ where ${\mathbb{F}}_q$ represents the field of q elements. Our study extends many of the results known for perturbations over the binary field to any finite field. In particular, we establish when a particular perturbation is asymptotically balanced over a prime field and provide a construction to find such perturbations over any finite field.

在本文中，我们建立了与基本对称函数的有限域上的指数和有关的生成函数的渐近行为及其摄动。这种渐近行为使我们能够计算度k的基本对称多项式及其扰动返回的概率的概率生成函数$\beta \in {\mathbb{F}}_q$ 哪里 ${\mathbb{F}}_q$表示q个元素的字段。我们的研究将对二进位场的扰动已知的许多结果扩展到任何有限场。特别是，我们确定何时在素数场上渐近平衡特定的扰动，并提供一种构造来在任何有限域上找到这种扰动。