Computing endomorphism rings of abelian varieties of dimension two

Generalizing a method of Sutherland and the author for elliptic curves, we design a subexponential algorithm for computing the endomorphism rings of ordinary abelian varieties of dimension two over finite fields. Although its correctness and complexity analysis rest on several assumptions, we report on practical computations showing that it performs very well and can easily handle previously intractable cases.Comment: 14 pages, 2 figure

A computable absolutely normal Liouville number

We give an algorithm that computes an absolutely normal Liouville number.Fil: Becher, Veronica Andrea. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Heiber, Pablo Ariel. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Slaman, Theodore A.. University of California at Berkeley; Estados Unido

Modular Las Vegas Algorithms for Polynomial Absolute Factorization

Let f(X,Y) \in \ZZ[X,Y] be an irreducible polynomial over \QQ. We give a Las Vegas absolute irreducibility test based on a property of the Newton polytope of $f$, or more precisely, of $f$ modulo some prime integer $p$. The same idea of choosing a $p$ satisfying some prescribed properties together with $LLL$ is used to provide a new strategy for absolute factorization of $f(X,Y)$. We present our approach in the bivariate case but the techniques extend to the multivariate case. Maple computations show that it is efficient and promising as we are able to factorize some polynomials of degree up to 400