Power maps and subvarieties of the complex algebraic $n$--torus

Given a subvariety $V$ of the complex algebraic torus ${\mathbb G}_{\rm m}^n$ defined by polynomials of total degree at most $d$ and a power map $\phi: {\mathbb G}_{\rm m}^n \to {\mathbb G}_{\rm m}^n$, the points ${\bf x}$ whose forward orbits ${\mathcal O}_\phi({\bf x})$ belong to $V$ form its {\em stable} subvariety $S(V,\phi)$. The main result of the paper provides an upper bound $T=T(n,d,\phi)$ for the number of iterations of the power map $\phi$ required to ``cut off'' the points of $V$ that do not belong to $S$.