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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....

Algorithms for quadratic forms over real function fields

Konrad Jałowiecki, Przemysław Koprowski (2016)

Banach Center Publications

This paper presents algorithms for quadratic forms over a formally real algebraic function field K of one variable over a fixed real closed field k. The algorithms introduced in the paper solve the following problems: test whether an element is a square, respectively a local square, compute Witt index of a quadratic form and test if a form is isotropic/hyperbolic. Finally, we remark on a method for testing whether two function fields are Witt equivalent.

Calcul du nombre de points sur une courbe elliptique dans un corps fini : aspects algorithmiques

François Morain (1995)

Journal de théorie des nombres de Bordeaux

Nous décrivons dans cet article les algorithmes nécessaires à une implantation efficace de la méthode de Schoof pour le calcul du nombre de points sur une courbe elliptique dans un corps fini. Nous tentons d’unifier pour cela les idées d’Atkin et d’Elkies. En particulier, nous décrivons le calcul d’équations pour X 0 ( ℓ ) , ℓ premier, ainsi que le calcul efficace de facteurs des polynômes de division d’une courbe elliptique.

Computations with Witt vectors of length 3

Luís R. A. Finotti (2011)

Journal de Théorie des Nombres de Bordeaux

In this paper we describe how to perform computations with Witt vectors of length 3 in an efficient way and give a formula that allows us to compute the third coordinate of the Greenberg transform of a polynomial directly. We apply these results to obtain information on the third coordinate of the j -invariant of the canonical lifting as a function on the j -invariant of the ordinary elliptic curve in characteristic p .

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