Linear equations with unknowns from a multiplicative group of finite rank whose solutions lie in a few subspaces
by
Jan-Hendrik Evertse
Universiteit Leiden, Mathematisch Instituut, Leiden, The Netherlands

Let G be a multiplicative subgroup of finite rank in a field K of characteristic 0. We consider equations (*) a₁x₁+ ... +a_nx_n=1 in unknowns x₁, ... , x_n in G, where a₁, ... , a_n are non-zero elements of K. Two tuples of coefficients (a₁, ... , a_n), (b₁, ... , b_n) are called equivalent if a_i/b_i in G for i=1, ... , n. Györy and the author proved in 1988 that for all tuples (a₁, ... , a_n), except for those lying in a finite number of equivalence classes, the set of solutions of (*) is contained in the union of at most 2^n! proper linear subspaces of Kⁿ. In 1992, the was improved by the author to (n!)²ⁿ⁺². We recently obtained the further improvement 2ⁿ⁺¹. In our lecture we will sketch the proof of this result.

Date received: April 28, 2004