$\G_m^{k}$.

Torsion points on hypersurfaces
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Results regarding uniform bounds on the number of torsion points in subvarieties of $\G_m^k$ go back to work of Bombieri and Zannier [@BZ95], Schlickewei [@Schl] and Evertse [@Ever]. For example, Evertse [@Ever], improving bounds of Schlickewei [@Schl], shows that the number of non-degenerate solutions in roots of unity to the equation $a_1x_1+\cdots+a_kx_k=1$, $a_1,\ldots,a_k\in\C$, is at most $(k+1)^{3(k+1)^2}$.

For our results we use the following result of Granville and Rudnick, see [@GranRud Corollary 3.1], which describes the structure of the algebraic subgroups that contain the roots of unity on a hypersurface. Although the statement of their result does not contain the bound for the degree or the number of the polynomials defining the algebraic subgroups, this follows directly from or is explicitly stated in their proof. Moreover, we recall this result only for the case of hypersurfaces, however their result holds for any algebraic variety.

\[lem:GranRud\] Let $\cH=Z(H)$ be a hypersurface in $\C^k$ defined by a polynomial $H\in\C[X_1,\ldots,X_k]$ of degree $D$ and with $s(H)$ terms. There exists a finite list $\cB$ of at most $$N(H) \le  (0.792s(H)/\log \(s(H)+1\))^{s(H)}$$ integer $k\times k$ matrices $B=(b_{j,i})$, $i,j=1,\ldots,k$, and $$b_{j,i}\le D\prod_{p\le s(H)}p,$$ where the product runs over all primes 