25,484 research outputs found
On the value set of small families of polynomials over a finite field, I
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 holds without
restrictions on the characteristic of Fq and asserts that
V(d,s,\bfs{a})=\mu_d.q+\mathcal{O}(1), where 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},..,
d_{d-s}). We provide an explicit upper bound for the constant underlying the
\mathcal{O}--notation in terms of d and s with "good" behavior. Our approach
reduces the question to estimate the number of Fq--rational points with
pairwise--distinct coordinates of a certain family of complete intersections
defined over Fq. We show that the polynomials defining such complete
intersections are invariant under the action of the symmetric group of
permutations of the coordinates. This allows us to obtain critical information
concerning the singular locus of the varieties under consideration, from which
a suitable estimate on the number of Fq--rational points is established.Comment: 30 page
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
Encoding points on hyperelliptic curves over finite fields in deterministic polynomial time
We present families of (hyper)elliptic curve which admit an efficient
deterministic encoding function
More Discriminants with the Brezing-Weng Method
The Brezing-Weng method is a general framework to generate families of
pairing-friendly elliptic curves. Here, we introduce an improvement which can
be used to generate more curves with larger discriminants. Apart from the
number of curves this yields, it provides an easy way to avoid endomorphism
rings with small class number
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