lying in the intersection of a curve $\cC$, not contained in any translate of a proper subtorus of $\G_m^{\ell}$, with the union of all proper algebraic subgroups is of bounded height (see also [@BMZ99 Theorem 1’]).

Furthermore, this also follows from previous work of N' eron [@Ne] (see also [@Serre Chapter 11]), Silverman [@Si83 Theorem C] and Masser [@Ma89] (see also [@Za Notes to Chapter 1] where Masser’s method is explained) on specialisations of finitely generated subgroups of abelian varieties. In particular, Masser’s result [@Ma89] gives explicit bounds for the least degree of a hypersurface containing the set of exceptional points, that is, points that lead to multiplicative dependence, of bounded degree and height.

Although the above results are sufficient for our purpose, for the sake of completeness we now give a simple self-contained proof that follows directly from Hilbert’s irreducibility theorem, see [@Sch Theorem 46]. Moreover, this proof does not appeal to the notion of height and applies to arbitrary Hilbertian field (see Definition \[def:HIT\] below), rather than to just finite extensions of $\Q$.

\[def:HIT\] We say that a field $\K$ is Hilbertian if for any irreducible polynomials $P_1,\ldots,P_r\in\K[X_1,\ldots,X_{\ell}]$ over $\K$ there exists a specialisation $(\alpha_2,\ldots,\alpha_{\ell})\in\K^{{\ell}-1}$ such that $P_i(X_1,\alpha_2,\ldots,\alpha_{\ell})$, 