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

We obtain an estimate on the average cardinality of the value set of any family of monic polynomials of Fq[T] of degree d for which s consecutive coefficients a_{d-1},…,a_{d-s} are fixed. Our estimate asserts that $\mathcal{V}$(d,s,\bfs{a})=\mu_d,q+$\mathcal{O}$(q^{1/2}), where $\mathcal{V}$(d,s,\bfs{a}) is such an average cardinality, \mu_d:=\sum_{r=1}^d{(-1)^{r-1}}/{r!} and \bfs{a}:=(a_{d-1},…,a_{d-s}). We also prove that $\mathcal{V}$_2(d,s,\bfs{a})=\mu_d^2,q^2+$\mathcal{O}$(q^{3/2}), where that $\mathcal{V}$_2(d,s,\bfs{a}) is the average second moment on any family of monic polynomials of Fq[T] of degree d with s consecutive coefficients fixed as above. Finally, we show that $\mathcal{V}$_2(d,0)=\mu_d^2,q^2+$\mathcal{O}$(q), where $\mathcal{V}$_2(d,0) denotes the average second moment of all monic polynomials in Fq[T] of degree d with f(0)=0. All our estimates hold for fields of characteristic p>2 and provide explicit upper bounds for the constants underlying the $\mathcal{O}$–notation in terms of d and s with “good” behavior. Our approach reduces the questions to estimate the number of Fq–rational points with pairwise–distinct coordinates of a certain family of complete intersections defined over Fq. A critical point for our results is an analysis of the singular locus of the varieties under consideration, which allows to obtain rather precise estimates on the corresponding number of Fq–rational points.