Searcharxiv⌕ Search

arXiv subjects

Enric Nart

Publications and source records attributed to Enric Nart.

At least 37 records · Page 2Linked to original sources
arXiv2019

Defectless polynomials over henselian fields and inductive valuations

Let $(K,v)$ be a henselian valued field. Let $\mathbb{P}^{dless}\subset K[x]$ be the set of monic, irreducible polynomials which are defectless and have degree greater than one. For a certain equivalence relation $\,\approx\,$ on $\,\mathbb{P}^{dless}$, we establish a canonical bijection $\mathbb{M}\to \mathbb{P}^{dless}/\!\!\approx$, where $\mathbb{M}$ is a discrete MacLane space, constructed in terms of inductive valuations on $K[x]$ extending $v$.

math.AG↗
arXiv2018

Key polynomials over valued fields

Let K be a field. For a given valuation on K[x], we determine the structure of its graded algebra and describe its set of key polynomials, in terms of any given key polynomial of minimal degree. We also characterize valuations not admitting key polynomials.

math.AG↗
arXiv2016

Reduced normal form of local integral bases

We introduce a canonical form for reduced bases of integral closures of discrete valuation rings, and we describe an algorithm for computing a basis in reduced normal form. This normal form has the same applications as the Hermite normal form: identification of isomorphic objects, construction of global bases by patching local ones, etc. but in addition the bases are reduced, which is a crutial property for several important applications. Except for very particular cases, a basis in Hermite normal form cannot be reduced.

math.NT↗
arXiv2015

Local-to-global computation of integral bases without a previous factorization of the discriminant

We adapt an old local-to-global technique of Ore to compute, under certain mild assumptions, an integral basis of a number field without a previous factorization of the discriminant of the defining polynomial. In a first phase, the method yields as a by-product successive splittings of the discriminant. When this phase concludes, it requires a squarefree factorization of some base factors of the discriminant to terminate.

math.NT↗
arXiv2015

On the equivalence of types

Types over a discrete valued field $(K,v)$ are computational objects that parameterize certain families of monic irreducible polynomials in $K_v[x]$, where $K_v$ is the completion of $K$ at $v$. Two types are considered to be equivalent if they encode the same family of prime polynomials. In this paper, we characterize the equivalence of types in terms of certain data supported by them.

math.NT↗
arXiv2014

Genetics of polynomials over local fields

Let $(K,v)$ be a discrete valued field with valuation ring $\oo$, and let $\oo_v$ be the completion of $\oo$ with respect to the $v$-adic topology. In this paper we discuss the advantages of manipulating polynomials in $\oo_v[x]$ in a computer by means of OM representations of prime (monic and irreducible) polynomials. An OM representation supports discrete data characterizing the Okutsu equivalence class of the prime polynomial. These discrete parameters are a kind of DNA sequence common to all individuals in the same Okutsu class, and they contain relevant arithmetic information about the polynomial and the extension of $K_v$ that it determines.

math.NT↗
arXiv2013

Residual ideals of MacLane valuations

Let $K$ be a field equipped with a discrete valuation $v$. In a pioneering work, S. MacLane determined all valuations on $K(x)$ extending $v$. His work was recently reviewed and generalized by M. Vaquié, by using the graded algebra of a valuation. We extend Vaquié's approach by studying residual ideals of the graded algebra of a valuation as an abstract counterpart of certain residual polynomials which play a key role in the computational applications of the theory. As a consequence, we determine the structure of the graded algebra of the discrete valuations on $K(x)$ and we show how these valuations may be used to parameterize irreducible polynomials over local fields up to Okutsu equivalence.

math.NT↗
arXiv2012

Local computation of differents and discriminants

We obtain several results on the computation of different and discriminant ideals of finite extensions of local fields. As an application, we deduce routines to compute the $\p$-adic valuation of the discriminant $\dsc(f)$, and the resultant $\res(f,g)$, for polynomials $f(x),g(x)\in A[x]$, where $A$ is a Dedekind domain and $\p$ is a non-zero prime ideal of $A$ with finite residue field. These routines do not require the computation of neither $\dsc(f)$ nor $\res(f,g)$; hence, they are useful in cases where this latter computation is inefficient because the polynomials have a large degree or very large coefficients.

math.NT↗
arXiv2012

Complexity of OM factorizations of polynomials over local fields

Let $k$ be a locally compact complete field with respect to a discrete valuation $v$. Let $\oo$ be the valuation ring, $\m$ the maximal ideal and $F(x)\in\oo[x]$ a monic separable polynomial of degree $n$. Let $δ=v(\dsc(F))$. The Montes algorithm computes an OM factorization of $F$. The single-factor lifting algorithm derives from this data a factorization of $F \md{\m^ν}$, for a prescribed precision $ν$. In this paper we find a new estimate for the complexity of the Montes algorithm, leading to an estimation of $O(n^{2+ε}+n^{1+ε}δ^{2+ε}+n^2ν^{1+ε})$ word operations for the complexity of the computation of a factorization of $F \md{\m^ν}$, assuming that the residue field of $k$ is small.

math.NT↗
arXiv2010

A new computational approach to ideal theory in number fields

Let $K$ be the number field determined by a monic irreducible polynomial $f(x)$ with integer coefficients. In previous papers we parameterized the prime ideals of $K$ in terms of certain invariants attached to Newton polygons of higher order of the defining equation $f(x)$. In this paper we show how to carry out the basic operations on fractional ideals of $K$ in terms of these constructive representations of the prime ideals. From a computational perspective, these results facilitate the manipulation of fractional ideals of $K$ avoiding two heavy tasks: the construction of the maximal order of $K$ and the factorization of the discriminant of $f(x)$. The main computational ingredient is Montes algorithm, which is an extremely fast procedure to construct the prime ideals.

math.NT↗
arXiv2010

Okutsu invariants and Newton polygons

Let K be a local field of characteristic zero, O its ring of integers and F(x) a monic irreducible polynomial with coefficients in O. K. Okutsu attached to F(x) certain primitive divisor polynomials F_1(x),..., F_r(x), that are specially close to F(x) with respect to their degree. In this paper we characterize the Okutsu families [F_1,..., F_r] in terms of certain Newton polygons of higher order, and we derive some applications: closed formulas for certain Okutsu invariants, the discovery of new Okutsu invariants, or the construction of Montes approximations to F(x); these are monic irreducible polynomials sufficiently close to F(x) to share all its Okutsu invariants. This perspective widens the scope of applications of Montes' algorithm, which can be reinterpreted as a tool to compute the Okutsu polynomials and a Montes approximation, for each irreducible factor of a monic separable polynomial f(x) in O[x].

math.NT↗
arXiv2009

Newton polygons and p-integral bases

Let p be a prime number. In this paper we use an old technique of Ore, based on Newton polygons, to construct in an efficient way p-integral bases of number fields defined by a p-regular equation. To illustrate the potential applications of this construction, we show how this result yields a computation of a p-integral basis of an arbitrary quartic field in terms of a defining equation.

math.NT↗
arXiv2009

Genus 3 curves with many involutions and application to maximal curves in characteristic 2

Let k=F_q be a finite field of characteristic 2. A genus 3 curve C/k has many involutions if the group of k-automorphisms admits a C_2\times C_2 subgroup H (not containing the hyperelliptic involution if C is hyperelliptic). Then C is an Artin-Schreier cover of the three elliptic curves obtained as the quotient of C by the nontrivial involutions of H, and the Jacobian of C is k-isogenous to the product of these three elliptic curves. In this paper we exhibit explicit models for genus 3 curves with many involutions, and we compute explicit equations for the elliptic quotients. We then characterize when a triple (E_1,E_2,E_3) of elliptic curves admits an Artin-Schreier cover by a genus 3 curve, and we apply this result to the construction of maximal curves. As a consequence, when q is nonsquare and m=\lfloor 2 sqrt(q) \rfloor = 1,5,7 mod 8, we obtain that N_q(3)=1+q+3m. We also show that this occurs for an infinite number of values of q nonsquare.

math.NT↗
arXiv2008

Higher Newton polygons in the computation of discriminants and prime ideal decomposition in number fields

We present an algorithm for computing discriminants and prime ideal decomposition in number fields. The algorithm is a refinement of a p-adic factorization method based on Newton polygons of higher order. The running-time and memory requirements of the algorithm appear to be very good: for a given prime number p, it computes the p-valuation of the discriminant and the factorization of p in a number field of degree 1000 in a few seconds, in a personal computer.

math.NT↗
arXiv2008

Newton polygons of higher order in algebraic number theory

We develop a theory of arithmetic Newton polygons of higher order, that provides the factorization of a separable polynomial over a $p$-adic field, together with relevant arithmetic information about the fields generated by the irreducible factors. This carries out a program suggested by Ø. Ore. As an application, we obtain fast algorithms to compute discriminants, prime ideal decomposition and integral bases of number fields.

math.NT↗
arXiv2007

Counting hyperelliptic curves that admit a Koblitz model

Let k be a finite field of odd characteristic. We find a closed formula for the number of k-isomorphism classes of pointed, and non-pointed, hyperelliptic curves of genus g over k, admitting a Koblitz model. These numbers are expressed as a polynomial in the cardinality q of k, with integer coefficients (for pointed curves) and rational coefficients (for non-pointed curves). The coefficients depend on g and the set of divisors of q-1 and q+1. These formulas show that the number of hyperelliptic curves of genus g suitable (in principle) of cryptographic applications is asymptotically (1-e^{-1})2q^{2g-1}, and not 2q^{2g-1} as it was believed. The curves of genus g=2 and g=3 are more resistant to the attacks to the DLP; for these values of g the number of curves is respectively (91/72)q^3+O(q^2) and (3641/2880)q^5+O(q^4).

math.NT↗
arXiv2007

Zeta function and cryptographic exponent of supersingular curves of genus 2

We compute in a direct (not algorithmic) way the zeta function of all supersingular curves of genus 2 over a finite field k, with many geometric automorphisms. We display these computations in an appendix where we select a family of representatives of all these curves up to geometric isomorphism and we exhibit equations and the zeta function of all their twists. As an application we obtain a direct computation of the cryptographic exponent of the Jacobians of these curves.

math.NT↗