given a semiabelian variety over a number field and a closed subvariety thereof, it is a theorem (of various people) that the torsion points on the subvariety have zariski closure equal to some finite union of torsion cosets of semiabelian subvarieties. we will focus on this question for (split) tori, where it manifests as the more concrete problem of finding solutions to a system of polynomial equations valued in roots of unity. we compare two effective methods for solving this problem: a combinatorial approach introduced by conway-jones, and a commutative algebra approach introduced by beukers-smyth.