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

Guillermo Matera, Mariana Pérez, and Melina Privitelli. "On the value set of small families of polynomials over a finite field, II." Acta Arithmetica 165.2 (2014): 141-179. <http://eudml.org/doc/279477>.

@article{GuillermoMatera2014,
abstract = {We obtain an estimate on the average cardinality (d,s,a) of the value set of any family of monic polynomials in $_q[T]$ of degree d for which s consecutive coefficients $a = (a_\{d-1\},...,a_\{d-s\})$ are fixed. Our estimate asserts that $(d,s,a) = μ_d q + (q^\{1/2\})$, where $μ_d := ∑_\{r=1\}^d ((-1)^\{r-1\})/(r!)$. We also prove that $₂(d,s,a) = μ²_d q² + (q^\{3/2\})$, where ₂(d,s,a) is the average second moment of the value set cardinalities for any family of monic polynomials of $_q[T]$ of degree d with s consecutive coefficients fixed as above. Finally, we show that $₂(d,0) = μ²_d q² + (q)$, where ₂(d,0) denotes the average second moment for all monic polynomials in $_q[T]$ 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 in terms of d and s with “good” behavior. Our approach reduces the questions to estimating the number of $_q$-rational points with pairwise distinct coordinates of a certain family of complete intersections defined over $_q$. 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 $_q$-rational points.},
author = {Guillermo Matera, Mariana Pérez, Melina Privitelli},
journal = {Acta Arithmetica},
keywords = {finite fields; average value set cardinality; average second moment; singular complete intersections; rational points},
language = {eng},
number = {2},
pages = {141-179},
title = {On the value set of small families of polynomials over a finite field, II},
url = {http://eudml.org/doc/279477},
volume = {165},
year = {2014},
}

TY - JOUR
AU - Guillermo Matera
AU - Mariana Pérez
AU - Melina Privitelli
TI - On the value set of small families of polynomials over a finite field, II
JO - Acta Arithmetica
PY - 2014
VL - 165
IS - 2
SP - 141
EP - 179
AB - We obtain an estimate on the average cardinality (d,s,a) of the value set of any family of monic polynomials in $_q[T]$ of degree d for which s consecutive coefficients $a = (a_{d-1},...,a_{d-s})$ are fixed. Our estimate asserts that $(d,s,a) = μ_d q + (q^{1/2})$, where $μ_d := ∑_{r=1}^d ((-1)^{r-1})/(r!)$. We also prove that $₂(d,s,a) = μ²_d q² + (q^{3/2})$, where ₂(d,s,a) is the average second moment of the value set cardinalities for any family of monic polynomials of $_q[T]$ of degree d with s consecutive coefficients fixed as above. Finally, we show that $₂(d,0) = μ²_d q² + (q)$, where ₂(d,0) denotes the average second moment for all monic polynomials in $_q[T]$ 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 in terms of d and s with “good” behavior. Our approach reduces the questions to estimating the number of $_q$-rational points with pairwise distinct coordinates of a certain family of complete intersections defined over $_q$. 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 $_q$-rational points.
LA - eng
KW - finite fields; average value set cardinality; average second moment; singular complete intersections; rational points
UR - http://eudml.org/doc/279477
ER -