We consider the complexity of a Lenstra Lenstra Lovász lattice reduction algorithm ([LLL]) in which the vectors are allowed to be linearly dependent and in which one also asks for the matrix of the transformation from the given generators to the reduced basis. The main problem will be to show that the entries of the transformation matrix remain bounded through the algorithm, with a reasonable b...

We have evaluated numerically the zeros of the partition function of the q-state Potts model on the square lattice with reduced interactions K . On the basis of our numerical results, we conjecture that, both for finite planar self-dual lattices and for lattices with free or periodic boundary conditions in the thermodynamic limit, the zeros in the Resxd . 0 region of the complex x ­ seK 2 1dypq...

We describe how a shortest vector of a 2-dimensional integral lattice corresponds to a best approximation of a unique rational number defined by the lattice. This rational number and its best approximations can be computed with the euclidean algorithm and its speedup by Schönhage (1971) from any basis of the lattice. The described correspondence allows, on the one hand, to reduce a basis of a 2...

The set {bi}i=1 is called basis of L, and is represented by a matrix B having as columns the bi’s. Each matrix of the form BP , with P ∈ GLn(Z), represents a basis of L. A lattice can have bases with arbitrarily long and skew vectors. See figure 1. By lattice basis reduction we mean solving the following problem: given a lattice basis formed by long (and probably skew) vectors, compute a basis ...