We obtain an estimate on the average cardinality of the value set of any family of monic polynomials of F q [T ] of degree d for which s consecutive coefficients a d−1 , . . . , a d−s are fixed. Our estimate holds without restrictions on the characteristic of F q and asserts thatis such an average cardinality, µ d := d r=1 (−1) r−1 /r! and a := (a d−1 , . . . , d d−s ). We provide an explicit upper bound for the constant underlying the O-notation in terms of d and s with "good" behavior. Our approach reduces the question to estimate the number of F q -rational points with pairwise-distinct coordinates of a certain family of complete intersections defined over F q . We show that the polynomials defining such complete intersections are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning the singular locus of the varieties under consideration, from which a suitable estimate on the number of F q -rational points is established.

We obtain estimates on the number |A λ | of elements on a linear family A of monic polynomials of Fq[T ] of degree n having factorization pattern λ := 1 λ 1 2 λ 2 · · · n λn . We show that |A λ | = T (λ) q n−m +O(q n−m−1/2 ), where T (λ) is the proportion of elements of the symmetric group of n elements with cycle pattern λ and m is the codimension of A. Furthermore, if the family A under consideration is "sparse", then |A λ | = T (λ) q n−m + O(q n−m−1 ). Our estimates hold for fields Fq of characteristic greater than 2. We provide explicit upper bounds for the constants underlying the O-notation in terms of λ and A with "good" behavior. Our approach reduces the question to estimate the number of Fq-rational points of certain families of complete intersections defined over Fq. Such complete intersections are defined by polynomials which are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning their singular locus, from which precise estimates on their number of Fq-rational points are established.

Abstract. We obtain an estimate on the average cardinality of the value set of any family of monic polynomials ofis the average second moment on any family of monic polynomials of F q [T ] of degree d with s consecutive coefficients fixed as above. Finally, we show that, where V 2 (d, 0) denotes the average second moment of all monic polynomials in F q [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 O-notation in terms of d and s with "good" behavior. Our approach reduces the questions to estimate the number of F q -rational points with pairwise-distinct coordinates of a certain family of complete intersections defined over F q . 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 F q -rational points.

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