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2014 | 165 | 2 | 141-179
Tytuł artykułu

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

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We obtain an estimate on the average cardinality 𝓥(d,s,a) of the value set of any family of monic polynomials in $𝔽_q[T]$ of degree d for which s consecutive coefficients $a = (a_{d-1},...,a_{d-s})$ are fixed. Our estimate asserts that $𝓥(d,s,a) = μ_d q + 𝓞(q^{1/2})$, where $μ_d := ∑_{r=1}^d ((-1)^{r-1})/(r!)$. We also prove that $𝓥₂(d,s,a) = μ²_d q² + 𝓞(q^{3/2})$, where 𝓥₂(d,s,a) is the average second moment of the value set cardinalities for any family of monic polynomials of $𝔽_q[T]$ of degree d with s consecutive coefficients fixed as above. Finally, we show that $𝓥₂(d,0) = μ²_d q² + 𝓞(q)$, where 𝓥₂(d,0) denotes the average second moment for all monic polynomials in $𝔽_q[T]$ of degree d with f(0) = 0. All our estimates hold for fields of characteristic p > 2 and provide explicit upper bounds for the 𝓞-constants in terms of d and s with "good" behavior. Our approach reduces the questions to estimating the number of $𝔽_q$-rational points with pairwise distinct coordinates of a certain family of complete intersections defined over $𝔽_q$. 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 $𝔽_q$-rational points.
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  • Instituto del Desarrollo Humano, Universidad Nacional, de General Sarmiento, J. M. Gutiérrez 1150, (B1613GSX) Los Polvorines, Buenos Aires, Argentina
  • National Council of Science, and Technology (CONICET), Argentina
  • Instituto del Desarrollo Humano, Universidad Nacional de General Sarmiento, J. M. Gutiérrez 1150, (B1613GSX) Los Polvorines, Buenos Aires, Argentina
  • Instituto de Ciencias, Universidad Nacional, de General Sarmiento, J. M. Gutiérrez 1150, (B1613GSX) Los Polvorines, Buenos Aires, Argentina
  • National Council of Science, and Technology (CONICET), Argentina
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bwmeta1.element.bwnjournal-article-doi-10_4064-aa165-2-3
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