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 $V_{f}={f(α)\midα\in\mathbb{F}{q}}$ 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.

💡 Research Summary

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The paper studies the size of the value set (V_f={f(\alpha)\mid \alpha\in\mathbb{F}_q}) of a polynomial (f\in\mathbb{F}_q