Algebraic Independence in Algebraic Groups. Part 1: Small Transcendence Degrees

Abstract

Our aim in this chapter is to provide an introduction, as simple as possible, to general statements of algebraic independence in the framework of algebraic groups. We consider only small transcendence degrees, by which we mean results asserting the algebraic independence of at least two numbers belonging to some set of numbers typically defined as values of the exponential map of some commutative algebraic group. Even in this special case, a single statement covering all the expected applications, remains to be settled. Here we have restricted our insight to a rather simple situation involving only a one parameter subgroup and for simplicity, we have also neglected the periods eventually contained in this subgroup. We give two statements which contain the most classical results, such as Gel’fond’s theorem.