On the Torelli problem and Jacobian Nullwerte in genus three

We give a closed formula for recovering a non-hyperelliptic genus three curve from its period matrix, and derive some identities between Jacobian Nullwerte in dimension three

Square-free OM computation of global integral bases

© 2022 Mathematical Sciences PublishersFor a prime p, the OM algorithm finds the p-adic factorization of an irreducible polynomial f¿Z[x]¿¿Z[¿] in polynomial time. This may be applied to construct p-integral bases in the number field K defined by f. In this paper, we adapt the OM techniques to work with a positive integer N instead of p. As an application, we obtain an algorithm to compute global integral bases in K, which does not require a previous factorization of the discriminant of f.Partially supported by grants MTM2015-66180-R and MTM2016-75980-P from the Spanish MECPeer ReviewedPostprint (author's final draft

Single-factor lifting and factorization of polynomials over local fields

Let f (x) be a separable polynomial over a local field. The Montes algorithm computes certain approximations to the different irreducible factors of f (x), with strong arithmetic properties. In this paper, we develop an algorithm to improve any one of these approximations, till a prescribed precision is attained. The most natural application of this ‘‘single-factor lifting’’ routine is to combine it with the Montes algorithm to provide a fast polynomial factorization algorithm. Moreover, the single-factor lifting algorithm may be applied as well to accelerate the computational resolution of several global arithmetic problems in which the improvement of an approximation to a single local irreducible factor of a polynomial is requiredPostprint (published version

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 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 idealsPostprint (author’s final draft

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 eld, together with relevant arithmetic information about the elds generated by the irreducible factors. This carries out a program suggested by . Ore. As an application, we obtain fast algorithms to compute discriminants, prime ideal decomposition and integral bases of number elds.Postprint (author’s final draft

Residual ideals of MacLane valuations

Let K be a field equipped with a discrete valuation v. In a pioneering work, MacLane determined all valuations on K(x) extending v. His work was recently reviewed and generalized by Vaqui´e, by using the graded algebra of a valuation. We extend Vaqui´e’s approach by studying residual ideals of the graded algebra 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 equivalencePostprint (author’s final draft

Valuative trees over valued fields

For an arbitrary valued field (K, v) and a given extension v(K*) ¿¿ ¿ of ordered groups, we analyze the structure of the tree formed by all ¿-valued extensions of v to the polynomial ring K[x]. As an application, we find a model for the tree of all equivalence classes of valuations on K[x] (without fixing their value group), whose restriction to K is equivalent to v.Peer ReviewedPostprint (author's final draft

Aprendre matemàtiques dissenyant

Matemàtiques per al disseny és una assignatura de l’EPSEVG (UPC). S’hi presenten els fonaments matemàtics del disseny industrial. Explicarem com hem incorporat la metodologia aprendre fent. A banda d’il·lustrar tots els conceptes amb objectes quotidians, basem l’aprenentatge en el treball per projectes. Això permet incorporar la creativitat dels estudiants i afavoreix la seva implicació. A més, hem usat les xarxes socials com a mitjà de comunicació.Postprint (published version

Lazlo Lovász i Avi Widergson: premis Abel 2021

El dia 17 de març l’Acadèmia de Ciències Noruega va anunciar que el Premi Abel 2021 s’atorgava a Laszló Lovász i Avi Widgerson per, segons es llegeix de la laudatio del premi, “. . . les seves contribucions fonamentals a la informàtica teòrica i les matemàtiques discretes, i el seu paper principal en la seva configuració en camps centrals de les matemàtiques modernes.Peer ReviewedPostprint (published version

SAGE, Matemàtiques interactives a l’abast

L'objectiu bàsic del projecte és augmentar significativament les capacitats d'autoaprenentatge dels estudiants de les assignatures de Matemàtiques del primer curs dels graus en enginyeria. El programari SAGE i els materials interactius creats haurien de cobrir aquest aspecte i alhora fer més atractiu l'estudi de les Matemàtiques als estudiants.Peer Reviewe