Linear equations in variables which Lie in a multiplicative group


Let $K$ be a field of characteristic $0$ and let $n$ be a natural number. Let $\Gamma$ be a subgroup of the multiplicative group $(K^\ast)^n$ of finite rank $r$. Given $a_1,\ldots,a_n\in K^\ast$ write $A(a_1,\ldots,a_n\Gamma)$ for the number of solutions $\mathrm{x} = (x_1,\ldots,x_n)\in\Gamma$ of the equation $a_1x_1 +\cdots+a_nx_n=1$, such that no proper subsum of $a_1x_1\cdots + a_nx_n$ vanishes. We derive an explicit upper bound for $A(a_1,\ldots,a_n,\Gamma)$ which depends only on the dimension $n$ and on the rank $r$.


Jan-Hendrik Evertse

Hans Peter Schlickewei

Wolfgang M. Schmidt