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

Abstract

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

12 pages, corrected typos