2013/10/11 by Guillermo Matera, Matera, Guillermo, Mariana Pérez +3
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1310.3293
openalex publication_date 2013/10/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ad-1,...,ad-s are fixed. Our estimate asserts that V(d,s,\bfsa)=μd q+O(q1/2), where V(d,s,\bfsa) is such an average cardinality, μd:=∑r=1d(-1)r-1/r! and \bfsa:=(ad-1,...,ad-s). We also prove that V2(d,s,\bfsa)=μd2 q2+O(q3/2), where that V2(d,s,\bfsa) 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 V2(d,0)=μd2 q2+O(q), where V2(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.