Linear equations with unknowns from a multiplicative group whose solutions lie in a small number of subspaces

AbstractLet K be a field of characteristic 0 and let (K*)n denote the n-fold Cartesian product of K*, endowed with coordinatewise multiplication. Let Γ be a subgroup of (K*)n of finite rank. We consider equations (*) a1x1 + … + anxn = 1 in x = (x1xn)Γ, where a = (a1,an)(K*)n. Two tuples a, b(K*)n are called Γ-equivalent if there is a uΓ such that b = u · a. Győry and the author [Compositio Math. 66 (1988) 329–354] showed that for all but finitely many Γ-equivalence classes of tuples a(K*)n, the set of solutions of (*) is contained in the union of not more than 2(n+1! proper linear subspaces of Kn. Later, this was improved by the author [J. reine angew. Math. 432 (1992) 177–217] to (n!)2n+2. In the present paper we will show that for all but finitely many Γ-equivalence classes of tuples of coefficients, the set of non-degenerate solutions of (*) (i.e., with non-vanishing subsums) is contained in the union of not more than 2n proper linear subspaces of Kn. Further we give an example showing that 2n cannot be replaced by a quantity smaller than n