Related papers: Using zeta functions to factor polynomials over fi…

Let $q$ be an odd prime power, and $H_{d,q}$ denote the set of square-free monic polynomials $D(x) \in F_q[x]$ of degree $d$. Katz and Sarnak showed that the moments, over $H_{d,q}$, of the zeta functions associated to the curves…

Combining the idea of motivic zeta function, due to Kapranov, and Pellikaan's definition of a two- variable zeta function for curves over finite fields in the present note we introduce a motivic two- variable zeta function for curves over…

In the present paper we provide a probabilistic polynomial time algorithm that reduces the complete factorization of any squarefree integer $n$ to counting points on elliptic curves modulo $n$, succeeding with probability $1-\varepsilon$,…

We propose and rigorously analyze two randomized algorithms to factor univariate polynomials over finite fields using rank $2$ Drinfeld modules. The first algorithm estimates the degree of an irreducible factor of a polynomial from…

The enumeration of points on (or off) the union of some linear or affine subspaces over a finite field is dealt with in combinatorics via the characteristic polynomial and in algebraic geometry via the zeta function. We discuss the basic…

Initially motivated by the relations between Anabelian Geometry and Artin's L-functions of the associated Galois-representations, here we study the list of zeta-functions of genus two abelian coverings of elliptic curves over finite fields.…

We present an efficient deterministic algorithm which outputs exact expressions in terms of $n$ for the number of monic degree $n$ irreducible polynomials over $\mathbb{F}_{q}$ of characteristic $p$ for which the first $l < p$ coefficients…

This paper provides an algorithm enumerating superspecial trigonal curves of genus $5$ over finite fields. Executing the algorithm over a computer algebra system Magma, we enumerate them over finite fields $\mathbb{F}_{p^a}$ for any natural…

The SEA algorithm for computing the cardinality of elliptic curves over finite fields in many characteristic uses modular polynomials. These polynomials come into different flavors, and methods to compute them flourished. Once equipped with…

We compute the number of points over finite fields of some algebraic varieties related to cluster algebras of finite type. More precisely, these varieties are the fibers of the projection map from the cluster variety to the affine space of…

We describe the practical implementation of an average polynomial-time algorithm for counting points on superelliptic curves defined over $\mathbb Q$ that is substantially faster than previous approaches. Our algorithm takes as input a…

We propose a Las Vegas probabilistic algorithm to compute the zeta function of a genus-3 hyperelliptic curve defined over a finite field $\mathbb F_q$, with explicit real multiplication by an order $\mathbb Z[\eta]$ in a totally real cubic…

In this work we relate the deterministic complexity of factoring polynomials (over finite fields) to certain combinatorial objects we call m-schemes. We extend the known conditional deterministic subexponential time polynomial factoring…

Stark and Terras introduced the edge zeta function of a finite graph in 1996. The edge zeta function is the reciprocal of a polynomial in twice as many variables as edges in the graph and can be computed in polynomial time. We look at graph…

We present a probabilistic Las Vegas algorithm for computing the local zeta function of a hyperelliptic curve of genus $g$ defined over $\mathbb{F}_q$. It is based on the approaches by Schoof and Pila combined with a modeling of the…

We study zeta functions enumerating subalgebras or ideals of Lie algebras over finite field of prime order $\mathbb{F}_p$. We first develop a general blueprint method for computing zeta functions of $\mathbb{F}_p$-Lie algebras, and…

We derive a new bound for some bilinear sums over points of an elliptic curve over a finite field. We use this bound to improve a series of previous results on various exponential sums and some arithmetic problems involving points on…

In this paper, we study square functions for extension operators over finite-type, planar curves endowed with the Euclidean arclength measure. We prove new results for curves of the form $(T,\phi(T))$ where $\phi(T)$ is a polynomial of…

Bhargava introduced a generalization of the factorial function to extend classical results in integers to Dedekind rings and unify them. We study the Bhargava factorial of the images of polynomial maps from an analytic perspective. We first…

We present a method for computing the zeta function of a smooth projective variety over a finite field which proceeds by induction on the dimension. We have implemented our approach for some surfaces using the Magma programming language,…