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

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 that V ( d , s , a ) = μ d q + O ( 1 ) , where V ( d , s , a ) is such an average cardinality, μ d : = ∑ r = 1 d ( − 1 ) r − 1 / r ! and a : = ( a d − 1 , … , a 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.