12 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
An approach to the moments subset sum problem through systems of diagonal equations over finite fields
Let be the finite field of elements, for a given subset
, , an integer and
we are interested in determining the
existence of a subset of cardinality such that for . This problem is known as the moment subset sum
problem and it is -complete for a general . We make a novel approach of
this problem trough algebraic geometry tools analyzing the underlying variety
and employing combinatorial techniques to estimate the number of
-rational points on certain varieties. We managed to give
estimates on the number of -rational points on certain diagonal
equations and use this results to give estimations and existence results for
the subset sum problem.Comment: 25 page
On the intrinsic complexity of the arithmetic Nullstellensatz
We show several arithmetic estimates for Hilbert's Nullstellensatz. This includes an algorithmic procedure computing the polynomials and constants occurring in a Bézout identity, whose complexity is polynomial in the geometric degree of the system. Moreover, we show for the first time height estimates of intrinsic type for the polynomials and constants appearing, again polynomial in the geometric degree and linear in the height of the system. These results are based on a suitable representation of polynomials by straight-line programs and duality techniques using the Trace Formula for Gorenstein algebras. As an application we show more precise upper bounds for the function πS(x) counting the number of primes yielding an inconsistent modular polynomial equation system. We also give a computationally interesting lower bound for the density of small prime numbers of controlled bit length for the reduction to positive characteristic of inconsistent systems. Again, this bound is given in terms of intrinsic parameters.Facultad de Ciencias Exacta
On the intrinsic complexity of the arithmetic Nullstellensatz
We show several arithmetic estimates for Hilbert's Nullstellensatz. This includes an algorithmic procedure computing the polynomials and constants occurring in a Bézout identity, whose complexity is polynomial in the geometric degree of the system. Moreover, we show for the first time height estimates of intrinsic type for the polynomials and constants appearing, again polynomial in the geometric degree and linear in the height of the system. These results are based on a suitable representation of polynomials by straight-line programs and duality techniques using the Trace Formula for Gorenstein algebras. As an application we show more precise upper bounds for the function πS(x) counting the number of primes yielding an inconsistent modular polynomial equation system. We also give a computationally interesting lower bound for the density of small prime numbers of controlled bit length for the reduction to positive characteristic of inconsistent systems. Again, this bound is given in terms of intrinsic parameters.Facultad de Ciencias Exacta
On the intrinsic complexity of the arithmetic Nullstellensatz
We show several arithmetic estimates for Hilbert's Nullstellensatz. This includes an algorithmic procedure computing the polynomials and constants occurring in a Bézout identity, whose complexity is polynomial in the geometric degree of the system. Moreover, we show for the first time height estimates of intrinsic type for the polynomials and constants appearing, again polynomial in the geometric degree and linear in the height of the system. These results are based on a suitable representation of polynomials by straight-line programs and duality techniques using the Trace Formula for Gorenstein algebras. As an application we show more precise upper bounds for the function πS(x) counting the number of primes yielding an inconsistent modular polynomial equation system. We also give a computationally interesting lower bound for the density of small prime numbers of controlled bit length for the reduction to positive characteristic of inconsistent systems. Again, this bound is given in terms of intrinsic parameters.Facultad de Ciencias Exacta