Perfect powers in value sets and orbits of polynomials

TL;DR

This paper proves the finiteness of perfect powers appearing in the orbits of polynomial dynamical systems over algebraic number fields, with results extended under the assumption of the abc-Conjecture.

Contribution

It establishes new finiteness results for perfect powers in polynomial orbits, including ratios, and connects these results to the abc-Conjecture.

Findings

Finiteness of perfect powers in polynomial orbits over algebraic number fields.

Finiteness of perfect powers in ratios of consecutive orbit elements.

Conditional finiteness results assuming the abc-Conjecture.

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.