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

2015/11/25 by Guillermo Matera, Matera, Guillermo, Mariana Pérez +3
Computer Science · Mathematics · #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1511.07942

openalex publication_date 2015/11/25 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

We estimate the average cardinality V(A) of the value set of a general family A of monic univariate polynomials of degree d with coefficients in the finite field \mathbbF_{\hskip-0.7mm q}. We establish conditions on the family A under which V(A)=μ_d q+O(q^1/2), where μ_d:=∑_r=1^d(-1)^r-1/r!. The result holds without any restriction on the characteristic of \mathbbF_{\hskip-0.7mm q} and provides an explicit expression for the constant underlying the O--notation in terms of d. We reduce the question to estimating the number of \mathbbF_{\hskip-0.7mm q}--rational points with pairwise--distinct coordinates of a certain family of complete intersections defined over \mathbbF_{\hskip-0.7mm q}. For this purpose, we obtain an upper bound on the dimension of the singular locus of the complete intersections under consideration, which allows us to estimate the corresponding number of \mathbbF_{\hskip-0.7mm q}--rational points.

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