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- W4300519772 abstract "We estimate the average cardinality $mathcal{V}(mathcal{A})$ of the value set of a general family $mathcal{A}$ of monic univariate polynomials of degree $d$ with coefficients in the finite field $mathbb{F}_{hskip-0.7mm q}$. We establish conditions on the family $mathcal{A}$ under which $mathcal{V}(mathcal{A})=mu_d,q+mathcal{O}(q^{1/2})$, where $mu_d:=sum_{r=1}^d{(-1)^{r-1}}/{r!}$. The result holds without any restriction on the characteristic of $mathbb{F}_{hskip-0.7mm q}$ and provides an explicit expression for the constant underlying the $mathcal{O}$--notation in terms of $d$. We reduce the question to estimating the number of $mathbb{F}_{hskip-0.7mm q}$--rational points with pairwise--distinct coordinates of a certain family of complete intersections defined over $mathbb{F}_{hskip-0.7mm q}$. For this purpose, we obtain an upper bound on the dimension of the singular locus of the complete intersections under consideration, which allows us to estimate the corresponding number of $mathbb{F}_{hskip-0.7mm q}$--rational points." @default.
- W4300519772 created "2022-10-03" @default.
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- W4300519772 date "2015-11-24" @default.
- W4300519772 modified "2023-10-14" @default.
- W4300519772 title "On the value set of small families of polynomials over a finite field, III" @default.
- W4300519772 doi "https://doi.org/10.48550/arxiv.1511.07942" @default.
- W4300519772 hasPublicationYear "2015" @default.
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