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

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 ${\mathbb{F}}_q$. We establish conditions on the family $\mathcal{A}$ under which $\mathcal{V}(\mathcal{A})={\mu }_{d}q+\mathcal{O}(q^{1\mathrm{/}2})$, where ${\mu }_d:={\sum }_{r=1}^d(-1{)}^{r-1}\mathrm{/}r!$. The result holds without any restriction on the characteristic of ${\mathbb{F}}_q$ 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 ${\mathbb{F}}_q$–rational points with pairwise–distinct coordinates of a certain family of complete intersections defined over ${\mathbb{F}}_q$. 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 ${\mathbb{F}}_q$–rational points.