Moreover, as we explain below, in certain situation we reduce our problem to applying [@AR Theorem 1], however for this we need a uniform bound for  that depends only on the degree of the polynomials $f$ and $g$.

Besides the generality of results, the new ingredients of the paper are employing results [@BS; @BMZ99; @BMZ08; @Mau] on the number of points on intersections of curves in the $n$-dimensional multiplicative torus $\G_m^n$ with algebraic subgroups. We also present two multivariate generalisations that are based on the use of Hilbert’s irreducibility theorem [@Sch] and a transformation using the Kronecker substitution to reduce the problem to the univariate case, as well as a result of Granville and Rudnick [@GranRud] about torsion points on hypersurfaces.

Conventions and notation
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As usual, we denote $\C[X_1,\ldots,X_{\ell}]$ the polynomial ring in ${\ell}$ variables and $\C(X_1,\ldots,X_{\ell})$ the field of rational functions $F/G$, $F,G\in\C[X_1,\ldots,X_{\ell}]$. When working with univariate polynomials we reserve the variable $T$. All polynomials in $\C[T]$ are denoted with small letters $f,g,\ldots$, and for polynomials in $\C[X_1,\ldots,X_\ell]$ we use capital letters $F,G,\ldots$.

Throughout the paper, for a univariate polynomial $f\in\C[T]$, the notation $d_f$ will be used for the degree of $f$.

For a family of polynomials $F_1,\ldots,F_s 