Open Diophantine Problems

We collect a number of open questions concerning Diophantine equations, Diophantine Approximation and transcendental numbers. Revised version: corrected typos and added references.Comment: 58 pages. to appear in the Moscow Mathematical Journal vo. 4 N.1 (2004) dedicated to Pierre Cartie

Extensions by Antiderivatives, Exponentials of Integrals and by Iterated Logarithms

Let F be a characteristic zero differential field with an algebraically closed field of constants, E be a no-new-constant extension of F by antiderivatives of F and let y1, ..., yn be antiderivatives of E. The antiderivatives y1, ..., yn of E are called J-I-E antiderivatives if the derivatives of yi in E satisfies certain conditions. We will discuss a new proof for the Kolchin-Ostrowski theorem and generalize this theorem for a tower of extensions by J-I-E antiderivatives and use this generalized version of the theorem to classify the finitely differentially generated subfields of this tower. In the process, we will show that the J-I-E antiderivatives are algebraically independent over the ground differential field. An example of a J-I-E tower is extensions by iterated logarithms. We will discuss the normality of extensions by iterated logarithms and produce an algorithm to compute its finitely differentially generated subfields.Comment: 66 pages, 1 figur

Dilogarithms, OPE and twisted T-duality

We study the full sigma model with target the three-dimensional Heisenberg nilmanifold by means of a Hamiltonian formulation of double field theory. We show that the expected T -duality with the sigma model on a torus endowed with H-flux is a manifest symmetry of the theory. We compute correlation functions of scalar fields and show that they exhibit dilogarithmic singularities. We show how the reflection and pentagonal identities of the dilogarithm can be interpreted in terms of correlators with 4 and 5 insertions.Comment: 33 page

Fast computation of power series solutions of systems of differential equations

We propose new algorithms for the computation of the first N terms of a vector (resp. a basis) of power series solutions of a linear system of differential equations at an ordinary point, using a number of arithmetic operations which is quasi-linear with respect to N. Similar results are also given in the non-linear case. This extends previous results obtained by Brent and Kung for scalar differential equations of order one and two

On computing real logarithms for matrices in the Lie group of special Euclidean motions in Rn

We show that the diagonal Pade approximants methods, both for computing the principal logarithm of matrices belonging to the Lie groupSE (n, IR) of special Euclidean motions in IRn and to compute the matrix exponential of elements in the corresponding Lie algebra se(n, IR), are structure preserving. Also, for the particular cases when n == 2,3 we present an alternative closed form to compute the principal logarithm. These low dimensional Lie groups play an important role in the kinematic motion of many mechanical systems and, for that reason, the results presented here have immediate applications in robotic