Related papers: Counting points on varieties over finite fields of…
We consider the problem of finding cryptographically suitable Jacobians. By applying a probabilistic generic algorithm to compute the zeta functions of low genus curves drawn from an arbitrary family, we can search for Jacobians containing…
We describe a method to accelerate the numerical computation of the coefficients of the polynomials $P_k(x)$ that appear in the conjectured asymptotics of the $2k$-th moment of the Riemann zeta function. We carried out our method to compute…
In 2010, A. Shpilka and I. Volkovich established a prominent result on the equivalence of polynomial factorization and identity testing. It follows from their result that a multilinear polynomial over the finite field of order 2 can be…
We establish a formula for the height zeta function for integral points on a class of projective toric varieties. Our method builds on the harmonic analysis approach of Batyrev--Tschinkel for rational points and is applicable even when the…
Given a sextic CM field $K$, we give an explicit method for finding all genus 3 hyperelliptic curves defined over $\mathbb{C}$ whose Jacobians are simple and have complex multiplication by the maximal order of this field, via an…
We give some bounds on the numbers of rational points on abelian varieties and jacobians varieties over finite fields. The main result is that we determine the maximum and minimum number of rational points on jacobians varieties of…
In this manuscript, we give effective methods for computing the zeta function of maximal orders on surfaces.
In this paper we give an algorithm to determine all finite matrix groups over a number field. Our algorithm is based on the representation theory of finite groups.
In this paper we present a p-adic algorithm to compute the zeta function of a nondegenerate curve over a finite field using Monsky-Washnitzer cohomology. The paper vastly generalizes previous work since all known cases, e.g. hyperelliptic,…
Many problems give rise to polynomial systems. These systems often have several parameters and we are interested to study how the solutions vary when we change the values for the parameters. Using predictor-corrector methods we track the…
We present a deterministic polynomial-time algorithm that determines whether a finite module over a finite commutative ring is cyclic, and if it is, outputs a generator.
We provide several properties of the geometric polynomials discussed in earlier works of the authors. Further, the geometric polynomials are used to obtain a closed form evaluation of certain series involving Riemann's zeta function.
Jacobi's $\theta$ function has numerous applications in mathematics and computer science; a naive algorithm allows the computation of $\theta(z,\tau)$, for $z, \tau$ verifying certain conditions, with precision $P$ in $O(\mathcal{M}(P)…
Let $Z$ be a projective geometrically integral algebraic variety. This paper is concerned with estimating the number of rational points on $Z$ which have height at most $B$. The bounds obtained are uniform in varieties of fixed degree and…
Given a projective variety X defined over a finite field, the zeta function of divisors attempts to count all irreducible, codimension one subvarieties of X, each measured by their projective degree. When the dimension of X is greater than…
We introduce a q-analogue of the classical Zeta polynomial of finite partially ordered sets, as a polynomial in one variable x with coefficients depending on the indeterminate q. We prove some properties of this polynomial invariant,…
We give deterministic polynomial-time algorithms that, given an order, compute the primitive idempotents and determine a set of generators for the group of roots of unity in the order. Also, we show that the discrete logarithm problem in…
A Hermite type formula is introduced and used to study the zeta function over the real and complex n-projective space. This approach allows to compute the residua at the poles and the value at the origin as well as the value of the…
Predicative analysis of recursion schema is a method to characterize complexity classes like the class FPTIME of polynomial time computable functions. This analysis comes from the works of Bellantoni and Cook, and Leivant by data tiering.…
We define the zeta function of a finite category. And we propose a conjecture which states the relationship between the Euler characteristic of finite categories and the zeta function of finite categories. This conjecture is verified when…