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

TL;DR

This paper estimates the average value set size of certain families of polynomials over finite fields, providing explicit bounds and reducing the problem to counting rational points on symmetric complete intersections.

Contribution

It introduces a new approach to estimate the value set size of polynomial families using algebraic geometry and symmetry properties, without restrictions on the field characteristic.

Findings

Average value set size approximates rac{d}{q} with explicit bounds.

Polynomials defining the varieties are invariant under permutation groups.

Provides explicit bounds for the error term in the estimate.

Abstract

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,\bfs{a})=\mu_d.q+\mathcal{O}(1), where V(d,s,\bfs{a}) is such an average cardinality, \mu_d:=\sum_{r=1}^d{(-1)^{r-1}}/{r!} and \bfs{a}:=(a_{d-1},.., d_{d-s}). We provide an explicit upper bound for the constant underlying the \mathcal{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 intersections 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 singular locus of the varieties under consideration, from which a suitable estimate on the number of Fq--rational points is established.