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

Tom 165 / 2014

Guillermo Matera, Mariana Pérez, Melina Privitelli Acta Arithmetica 165 (2014), 141-179 MSC: Primary 11T06; Secondary 11G25, 14B05, 14G05. DOI: 10.4064/aa165-2-3

Streszczenie

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.

Autorzy

  • Guillermo MateraInstituto 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
    e-mail
  • Mariana PérezInstituto del Desarrollo Humano
    Universidad Nacional de General Sarmiento
    J. M. Gutiérrez 1150
    (B1613GSX) Los Polvorines
    Buenos Aires, Argentina
    e-mail
  • Melina PrivitelliInstituto 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
    e-mail

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