Let $\mathbb{ F}_{ q}$ be a finite field of characteristic $p$, and let $f \in \mathbb{ F}_{ q}[x]$ be a polynomial of degree $d > 0$. Denote the image set of this polynomial as $V_{ f}=\{ f(\alpha)\mid\alpha\in\mathbb{ F}_{ q}\}$ and denote the cardinality of this set as $N_{ f}$. A much sharper bound for $N_{ f}$ is established in this paper. In particular, for any $p\neq 2, 3$, and for nearly every generic quartic polynomial $f \in \mathbb{ F}_{ q}[x]$, we obtain $$\lvert N_f - \frac{ 5}{ 8} q \rvert \leq \frac{ 1}{ 2}\sqrt{ q} + \frac{ 15}{ 4},$$ which holds as a simple corollary of the main result.
Categories: math.NT