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

TL;DR

This paper studies the behavior of points under power maps in complex algebraic tori, providing bounds on iterations needed to identify points whose orbits stay within a given subvariety.

Contribution

It introduces a bound on the number of power map iterations needed to distinguish points with orbits contained in a subvariety of the complex algebraic torus.

Findings

Established an explicit upper bound T(n,d,φ) for orbit containment.

Characterized stable subvarieties under power maps.

Enhanced understanding of dynamics in algebraic tori.

Abstract

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 $S$.