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An effective Bertini theorem and the number of rational points of a normal complete intersection over a finite field

100%

Cafure A., Matera G.

Acta Arithmetica

2007

tom 130

nr 1

19-35

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

81%

Matera G., Pérez M., Privitelli M.

Acta Arithmetica

2014

tom 165

nr 2

141-179

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.

Ograniczanie wyników

2 Acta Arithmetica

2 Matera G.

1 Cafure A.

1 Privitelli M.

1 Pérez M.

1 2014

1 2007

Wyniki wyszukiwania

An effective Bertini theorem and the number of rational points of a normal complete intersection over a finite field

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