Orbits under algebraic groups and logarithms of algebraic numbers

Simultaneous Approximation of Logarithms of Algebraic Numbers

Recently, close connections have been established between simultaneous diophantine approximation and algebraic independence. A survey of this topic is given by M. Laurent in these proceedings [7]. These connections are one of the main motivations to investigate systematically the question of algebraic approximation to transcendental numbers. In view of the applications to algebraic independence...

Quadratic relations between logarithms of algebraic numbers

A well known conjecture asserts that Q-linearly independent logarithms of algebraic numbers should be algebraically independent. So far it has only been proved that they are linearly independent over the field of algebraic numbers (Baker, 1966). It is not yet known that there exist two logarithms of algebraic numbers which are algebraically independent. The next step might be to consider quadra...

Orbits of algebraic numbers with low heights

We prove that the two smallest values of h(α)+h( 1 1−α )+h(1− 1 α ) are 0 and 0.4218 . . . , for α any algebraic integer. Introduction For K an algebraic number field, let Kv be the completion of K at the place v and let | |v be the absolute value associated with this completion Kv (more precise definitions are given below). For α ∈ K, we define the (logarithmic) Weil height, h(α), as follows:

On Closedness of Orbits of Algebraic Groups

Summarizing of Orbits of Algebraic Groups I

Let G̃ be a simple algebraic group which is defined and split over a field K and let L̃ be a corresponding Lie algebra. Further, let R be the corresponding root system. We prove here that if char K = 0 or a very good prime for R and | K |>| R |, then there exists an orbit O ⊂ L = L̃(K) with respect to the adjoint action of G = G̃(K) such that L = O + O. This is an analogue of the corresponding resu...