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

Abstract

We obtain an estimate on the average cardinality ν(d, s, a) of the value set of any family of monic polynomials in double-struck Fq[T] of degree d for which s consecutive coefficients a = (ad-1, ..., ad-s) are fixed. Our estimate asserts that, ν(d, s, a) = μdq + O(q1/2), where μd := Σr=1d(-1)r-1/r!. We also prove that ν2(d, s, a) = μd2q2 + O(q3/2), where ν2(d, s, a) is the average second moment of the value set cardinalities for any family of monic polynomials of double-struck Fq[T] of degree d with s consecutive coefficients fixed as above. Finally, we show that ν2(d, 0) = μd2q2 + O(q), where ν2(d, 0) denotes the average second moment for all monic polynomials in double-struck 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 O-constants in terms of d and s with "good" behavior. Our approach reduces the questions to estimating the number of double-struck Fq- rational points with pairwise distinct coordinates of a certain family of complete intersections defined over double-struck Fq. Critical to our results is the analysis of the singular locus of the varieties under consideration, which allows us obtain rather precise estimates on the corresponding number of double-struck Fq-rational points.