# Orbits of Polynomial Dynamical Systems Modulo Primes

## TL;DR

This paper establishes lower bounds on the orbit lengths of polynomial dynamical systems modulo primes, using recent theoretical results to identify families with long orbits and improving previous bounds.

## Contribution

It provides explicit bounds and families of polynomials with long orbits modulo primes, extending and refining earlier results in the field.

## Key findings

- Explicit families with long orbits modulo primes
- Improved lower bounds for orbit lengths
- Connections to previous results by Silverman and Akbary-Ghioca

## Abstract

We present lower bounds for the orbit length of reduction modulo primes of parametric polynomial dynamical systems defined over the integers, under a suitable hypothesis on its set of preperiodic points over $\mathbb C$. Applying recent results of Baker and DeMarco~(2011) and of Ghioca, Krieger, Nguyen and Ye~(2017), we obtain explicit families of parametric polynomials and initial points such that the reductions modulo primes have long orbits, for all but a finite number of values of the parameters. This generalizes a previous lower bound due to Chang~(2015). As a by-product, we also slighly improve a result of Silverman~(2008) and recover a result of Akbary and Ghioca~(2009) as special extreme cases of our estimates.