Linear Equations with Unknowns from a Multiplicative Group Whose Solutions Lie in a Small Number of Subspaces

Abstract. Let K be a field of characteristic 0 and let (K) denote the n-fold cartesian product of K∗, endowed with coordinatewise multiplication. Let Γ be a subgroup of (K) of finite rank. We consider equations (*) a1x1+ · · ·+anxn = 1 in x = (x1, . . . , xn) ∈ Γ, where a = (a1, . . . , an) ∈ (K). Two tuples a,b ∈ (K) are called Γ-equivalent if there is a u ∈ Γ such that b = u · a. Győry and the author [4] showed that for all but finitely many Γ-equivalence classes of tuples a ∈ (K), the set of solutions of (*) is contained in the union of not more than 2 proper linear subspaces of K. Later, this was improved by the author [3] to (n!). 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 2 proper linear subspaces of K. Further we give an example showing that 2 cannot be replaced by a quantity smaller than n.