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We obtain an estimate on the average cardinality of the value set of anyfamily of monic polynomials of Fq[T] of degree d for which s consecutivecoefficients a_{d-1},..., a_{d-s} are fixed. Our estimate holds withoutrestrictions on the characteristic of Fq and asserts thatV(d,s,\bfs{a})=\mu_d.q+\mathcal{O}(1), where V(d,s,\bfs{a}) is such an averagecardinality, \mu_d:=\sum_{r=1}^d{(-1)^{r-1}}/{r!} and \bfs{a}:=(a_{d-1},..,d_{d-s}). We provide an explicit upper bound for the constant underlying the\mathcal{O}--notation in terms of d and s with "good" behavior. Our approachreduces the question to estimate the number of Fq--rational points withpairwise--distinct coordinates of a certain family of complete intersectionsdefined over Fq. We show that the polynomials defining such completeintersections are invariant under the action of the symmetric group ofpermutations of the coordinates. This allows us to obtain critical informationconcerning the singular locus of the varieties under consideration, from whicha suitable estimate on the number of Fq--rational points is established.

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