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3-Selmer groups for curves y 2 = x 3 + a

Andrea Bandini (2008)

Czechoslovak Mathematical Journal

We explicitly perform some steps of a 3-descent algorithm for the curves y 2 = x 3 + a , a a nonzero integer. In general this will enable us to bound the order of the 3-Selmer group of such curves.

A computer algorithm for finding new euclidean number fields

Roland Quême (1998)

Journal de théorie des nombres de Bordeaux

This article describes a computer algorithm which exhibits a sufficient condition for a number field to be euclidean for the norm. In the survey [3] p 405, Franz Lemmermeyer pointed out that 743 number fields where known (march 1994) to be euclidean (the first one, ℚ , discovered by Euclid, three centuries B.C.!). In the first months of 1997, we found more than 1200 new euclidean number fields of degree 4, 5 and 6 with a computer algorithm involving classical lattice properties of the embedding of...

A fast algorithm for polynomial factorization over ℚ p

David Ford, Sebastian Pauli, Xavier-François Roblot (2002)

Journal de théorie des nombres de Bordeaux

We present an algorithm that returns a proper factor of a polynomial Φ ( x ) over the p -adic integers ℤ p (if Φ ( x ) is reducible over ℚ p ) or returns a power basis of the ring of integers of ℚ p [ x ] / Φ ( x ) ℚ p [ x ] (if Φ ( x ) is irreducible over ℚ p ). Our algorithm is based on the Round Four maximal order algorithm. Experimental results show that the new algorithm is considerably faster than the Round Four algorithm.

A generalization of the LLL-algorithm over euclidean rings or orders

Huguette Napias (1996)

Journal de théorie des nombres de Bordeaux

Numerous important lattices ( D 4 , E 8 , the Coxeter-Todd lattice K 12 , the Barnes-Wall lattice Λ 16 , the Leech lattice Λ 24 , as well as the 2 -modular 32 -dimensional lattices found by Quebbemann and Bachoc) possess algebraic structures over various Euclidean rings, e.g. Eisenstein integers or Hurwitz quaternions. One obtains efficient algorithms by performing within this frame the usual reduction procedures, including the well known LLL-algorithm.

A local limit theorem with speed of convergence for euclidean algorithms and diophantine costs

Viviane Baladi, Aïcha Hachemi (2008)

Annales de l'I.H.P. Probabilités et statistiques

For large N, we consider the ordinary continued fraction of x=p/q with 1≤p≤q≤N, or, equivalently, Euclid’s gcd algorithm for two integers 1≤p≤q≤N, putting the uniform distribution on the set of p and qs. We study the distribution of the total cost of execution of the algorithm for an additive cost function c on the set ℤ+* of possible digits, asymptotically for N→∞. If c is nonlattice and satisfies mild growth conditions, the local limit theorem was proved previously by the second named author....

A new interpretor for PARI/GP

Bill Allombert (2008)

Journal de Théorie des Nombres de Bordeaux

When Henri Cohen and his coworkers set out to write PARI twenty years ago, GP was an afterthought. While GP has become the most commonly used interface to the PARI library by a large margin, both the gp interpretor and the GP language are primitive in design. Paradoxically, while gp allows to handle very high-level objects, GP itself is a low-level language coming straight from the seventies.We rewrote GP as a compiler/evaluator pair, implementing several high-level features (statically scoped variables,...

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