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

Guillermo Matera ; Mariana Pérez ; Melina Privitelli

Acta Arithmetica, Tome 166 (2014), p. 141-179 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2014-01-01

Zbl 06345247

EUDML-ID : urn:eudml:doc:279477