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

Guillermo Matera¹ ; Mariana Pérez² ; Melina Privitelli³

¹ Instituto del Desarrollo Humano Universidad Nacional de General Sarmiento J. M. Gutiérrez 1150 (B1613GSX) Los Polvorines Buenos Aires, Argentina and National Council of Science and Technology (CONICET), Argentina
² Instituto del Desarrollo Humano Universidad Nacional de General Sarmiento J. M. Gutiérrez 1150 (B1613GSX) Los Polvorines Buenos Aires, Argentina
³ Instituto de Ciencias Universidad Nacional de General Sarmiento J. M. Gutiérrez 1150 (B1613GSX) Los Polvorines Buenos Aires, Argentina and National Council of Science and Technology (CONICET), Argentina

Acta Arithmetica, Tome 165 (2014) no. 2, pp. 141-179

Résumé

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.

Détail

DOI : 10.4064/aa165-2-3

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