Improving bounds for value sets of polynomials over finite fields

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 $p\mathrm{\ne }2,3$, and for nearly every generic quartic polynomial $f\in {\mathbb{F}}_q[x]$, we obtain $$\mid N_f-\frac{5}{8}q\mid \le \frac{1}{2}\sqrt{q}+\frac{15}{4},$$ which holds as a simple corollary of the main result.