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

CESARATTO, EDA; MATERA, GUILLERMO; PÉREZ, MARIANA; PRIVITELLI, MELINA

JOURNAL OF COMBINATORIAL THEORY SERIES A

ACADEMIC PRESS INC ELSEVIER SCIENCE

Lugar: Amsterdam; Año: 2014 vol. 124 p. 203 - 227

0097-3165

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 holds without restrictions on the characteristic of Fq and asserts that V(d, s, a) = μ_d q + O(1), where V(d, s, a) is such an average cardinality, μd equals the first d terms in the expansion of 1-exp(-1) 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 Fq -rational points with pairwise-distinct coordinates of a certain family of complete intersec- tions defined over Fq . 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 singu- lar locus of the varieties under consideration, from which a suitable estimate on the number of Fq -rational points is es- tablished.