Improving bounds for value sets of polynomials over finite fields

TL;DR

This paper derives sharper bounds for the size of value sets of polynomials over finite fields, especially for generic quartic polynomials, improving understanding of polynomial image distributions.

Contribution

It establishes new, tighter bounds for the cardinality of polynomial value sets over finite fields, particularly for quartic polynomials, advancing previous bounds significantly.

Findings

For generic quartic polynomials, the value set size is close to 5/8 of the field size.

The bounds hold for all characteristics except 2 and 3.

A specific inequality relates the value set size to the field size with a small error term.

Abstract

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.