Jordi Guàrdia

Jacobian nullwerte and algebraic equations

Abstract We present two applications of jacobian nullwerte, both related with the resolution of algebraic equations of any degree. We give a very simple expression of the roots of a polynomial of arbitrary degree in terms of derivatives of hyperelliptic theta functions. This expression can be understood as an explicit proof of Torellis theorem in the hyperelliptic case. We also give geometrical expressions of the discriminant of a polynomial. Both applications are based on a jacobian version of Thomaes formula.

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 the Montes algorithm, which is an extremely fast procedure to construct the prime ideals.

Trends in Number Theory

We show the existence of families of elliptic curves over Q whose generic rank is at least 2 for the torsion groups Z/8Z and Z/2Z x Z/6Z. Also in both cases we prove the existence of infinitely many elliptic curves, which are parameterized by the points of an elliptic curve with positive rank, with such torsion group and rank at least 3. These results represent an improvement of previous results by Campbell, Kulesz, Lecacheux, Dujella and Rabarison where families with rank at least 1 were constructed in both cases.