Value distribution of elementary symmetric polynomials and its perturbations over finite fields

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 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.