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An upper bound on the number of rational points of arbitrary projective varieties over finite fields
We give an upper bound on the number of rational points of an arbitrary
Zariski closed subset of a projective space over a finite field. This bound
depends only on the dimensions and degrees of the irreducible components and
holds for very general varieties, even reducible and non equidimensional. As a
consequence, we prove a conjecture of Ghorpade and Lachaud on the maximal
number of rational points of an equidimensional projective variety
On the value set of small families of polynomials over a finite field, II
We obtain an estimate on the average cardinality of the value set of any
family of monic polynomials of Fq[T] of degree d for which s consecutive
coefficients a_{d-1},...,a_{d-s} are fixed. Our estimate asserts that
\mathcal{V}(d,s,\bfs{a})=\mu_d\,q+\mathcal{O}(q^{1/2}), where
\mathcal{V}(d,s,\bfs{a}) is such an average cardinality,
\mu_d:=\sum_{r=1}^d{(-1)^{r-1}}/{r!} and \bfs{a}:=(a_{d-1},...,a_{d-s}). We
also prove that \mathcal{V}_2(d,s,\bfs{a})=\mu_d^2\,q^2+\mathcal{O}(q^{3/2}),
where that \mathcal{V}_2(d,s,\bfs{a}) is the average second moment on any
family of monic polynomials of Fq[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 of all monic polynomials in Fq[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 constants underlying the
\mathcal{O}--notation in terms of d and s with "good" behavior. Our approach
reduces the questions to estimate the number of Fq--rational points with
pairwise--distinct coordinates of a certain family of complete intersections
defined over Fq. A critical point for our results is an analysis of the
singular locus of the varieties under consideration, which allows to obtain
rather precise estimates on the corresponding number of Fq--rational points.Comment: 36 page
Logarithmic forms and singular projective foliations
In this article we study polynomial logarithmic -forms on a projective
space and characterize those that define singular foliations of codimension
. Our main result is the algebraic proof of their infinitesimal stability
when with some extra degree assumptions. We determine new irreducible
components of the moduli space of codimension two singular projective
foliations of any degree, and we show that they are generically reduced in
their natural scheme structure. Our method is based on an explicit description
of the Zariski tangent space of the corresponding moduli space at a given
generic logarithmic form. Furthermore, we lay the groundwork for an extension
of our stability results to the general case .Comment: Version 3. 29 pages. Some grammar mistakes and typos were fixed. This
article will appear at Annales de l'Institut Fourie
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