Perfect powers in value sets and orbits of polynomials

Abstract

We show the finiteness of perfect powers in orbits of polynomial dynamical systems over an algebraic number field. We also obtain similar results for perfect powers represented by ratios of consecutive elements in orbits. Assuming the $abc$-Conjecture for number fields, we obtain a finiteness result for powers in ratios of arbitrary elements in orbits.

arXiv commentIn this version we fixed a few inaccuracies, and heavily simplified the proof of Theorem 1.2. This version is accepted for publication in the New York Journal of Mathematics

References in corpus (1)

• Avoiding algebraic integers of bounded house in orbits of rational functions over cyclotomic closures

Cited by in corpus (1)

• Effective bounds on multiplicatively dependent orbits of integer polynomials modulo S-integers