On the value set of small families of polynomials over a finite field, II
Volume 165 / 2014
Abstract
We obtain an estimate on the average cardinality $\mathcal{V}(d,s,\boldsymbol{a})$ of the value set of any family of monic polynomials in $\mathbb F_q[T]$ of degree $d$ for which $s$ consecutive coefficients $\boldsymbol{a} = (a_{d-1},\dots, a_{d-s})$ are fixed. Our estimate asserts that $\mathcal{V}(d,s,\boldsymbol{a})=\mu_d q+\mathcal{O}(q^{{1}/{2}})$, where $\mu_d:=\sum_{r=1}^d{(-1)^{r-1}}/{r!}$. We also prove that $\mathcal{V}_2(d,s,\boldsymbol{a})=\mu_d^2 q^2+\mathcal{O}(q^{{3}/{2}})$, where $\mathcal{V}_2(d,s,\boldsymbol{a})$ is the average second moment of the value set cardinalities for any family of monic polynomials of $\mathbb F_q[T]$ of degree $d$ with $s$ consecutive coefficients fixed as above. Finally, we show that $\mathcal{V}_2(d,0)=\mu_d^2 q^2+\mathcal{O}(q)$, where $\mathcal{V}_2(d,0)$ denotes the average second moment for all monic polynomials in $\mathbb F_q[T]$ of degree $d$ with $f(0)=0$. All our estimates hold for fields of characteristic $p>2$ and provide explicit upper bounds for the $\mathcal{O}$-constants in terms of $d$ and $s$ with “good” behavior. Our approach reduces the questions to estimating the number of $\mathbb F_q$-rational points with pairwise distinct coordinates of a certain family of complete intersections defined over $\mathbb F_q$. 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 $\mathbb F_q$-rational points.
