ON THE VALUE SET OF SMALL FAMILIES OF POLYNOMIALS OVER A FINITE FIELD, III

GUILLERMO MATERA; MARIANA PÉREZ; MELINA PRIVITELLI

Contemporary developments in finite fields and applications

World Scientific

Año: 2016; p. 217 - 243

We obtain an estimate on the average cardinality of the value set on a linearfamily A of monic polynomials of Fq[T] of degree d. Our estimate asserts that V(A) =µd q + O(q1/2), where V(A) is such an average cardinality and µd := Pdr(−1)r−1/r!.The result holds for fields of characteristic p > 2 and provide explicit upper bounds for theconstants underlying the O-notation in terms of d with ?good? behavior. We reduce thequestion to estimate the number of Fq-rational points with pairwise-distinct coordinates of a certain family of complete intersections defined over Fq. For this purpose, we obtainan upper bound on the dimension of the singular locus of the complete intersections underconsideration, which allows us to estimate the corresponding number of Fq-rational points.