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

GUILLERMO MATERA; MARIANA PÉREZ; MELINA PRIVITELLI

ACTA ARITHMETICA

POLISH ACAD SCIENCES INST MATHEMATICS

Lugar: VARSOVIA; Año: 2014 vol. 165 p. 141 - 179

0065-1036

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 asserts that V(d,s,a)=mu_d q+O(q^{1/2}), where V(d,s,a) is such an average cardinality, mu_d:=sum_{r=1}^d (-1)^{r-1}/r! and a:=(a_{d-1},...,a_{d-s}). We also prove that V_2(d,s,a)=mu_d^2 q^2+O(q^{3/2}), where V_2(d,s,a) is the average second moment on any family of monic polynomials of Fq[T] of degree d with s consecutive coefficients fixed as above. Finally, we show that V_2(d,0)=mu_d^2 q^2+O(q), where V_2(d,0) denotes the average second moment of all monic polynomials in Fq[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 Fq-rational points with pairwise-distinct coordinates of a certain family of complete intersections defined over Fq. 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 Fq-rational points.