On the value set of small families of polynomials over a finite field, III

TL;DR

This paper estimates the average size of the value set of certain polynomial families over finite fields, providing explicit formulas and conditions, and employs algebraic geometry techniques to count rational points on related varieties.

Contribution

It establishes general conditions for the value set size of polynomial families over finite fields and derives explicit bounds using algebraic geometry methods.

Findings

Average value set size approximates rac{d}{q} with explicit constants.

Provides bounds on the number of rational points on specific algebraic varieties.

No restrictions on the characteristic of the finite field.

Abstract

We estimate the average cardinality $\mathcal{V}(\mathcal{A})$ of the value set of a general family $\mathcal{A}$ of monic univariate polynomials of degree $d$ with coefficients in the finite field $\mathbb{F}_{\hskip-0.7mm q}$. We establish conditions on the family $\mathcal{A}$ under which $\mathcal{V}(\mathcal{A})=\mu_d\,q+\mathcal{O}(q^{1/2})$, where $\mu_d:=\sum_{r=1}^d{(-1)^{r-1}}/{r!}$. The result holds without any restriction on the characteristic of $\mathbb{F}_{\hskip-0.7mm q}$ and provides an explicit expression for the constant underlying the $\mathcal{O}$--notation in terms of $d$. We reduce the question to estimating the number of $\mathbb{F}_{\hskip-0.7mm q}$--rational points with pairwise--distinct coordinates of a certain family of complete intersections defined over $\mathbb{F}_{\hskip-0.7mm q}$. For this purpose, we obtain an upper bound on the dimension of the singular locus of the complete intersections under consideration, which allows us to estimate the corresponding number of $\mathbb{F}_{\hskip-0.7mm q}$--rational points.