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. We consider linear equations a1*x1+...+an*xn=1 in unknowns x1,...,xn from G, where a1,...,an are non-zero elements of K, and where G is a subgroup of the multiplicative group of non-zero elements of K. Two tuples (a1,...,an) and (b1,...,bn) of non-zero elements of K are called G-equivalent if there are u1,...,un in G such that b1=a1*u1,..., bn=an*un. Denote by m(a1,...,an,G) the smallest number m such that the set of solutions of a1*x1+...+an*xn=1 in x1,...,xn from G is contained in the union of m proper linear subspaces of K^n. It is known that m(a1,...,an,G) is finite; clearly, this quantity does not change if (a1,...,an) is replaced by a G-equivalent tuple. Gyory and the author proved in 1988 that there is a constant c(n) depending only on the number of variables n, such that for all but finitely many G-equivalence classes (a1,...,an), one has m(a1,...,an,G)< c(n). It is as yet not clear what is the best possible value of c(n). Gyory and the author showed that c(n)=2^{(n+1)!} can be taken. This was improved by the author in 1993 to c(n)=(n!)^{2n+2}. In the present paper we improve this further to c(n)=2^{n+1}, and give an example showing that c(n) can not be smaller than n.