Subfield value sets of polynomials over finite fields

Abstract

Let $\mathbb{F}_{q^e}$ be a finite field, and let $\mathbb{F}_{q^d}$ be a subfield of $\mathbb{F}_{q^e}$. The \emph{value set} of a polynomial $f$ lying within $\mathbb{F}_{q^d}$ is defined as the set of images $\{f(c) \in \\mathbb{F}_{q^d}\colon c \in \mathbb{F}_{q^e}\}$. This work is concerned with the cardinality of value sets of polynomials lying within subfields.