Dynamically distinguishing polynomials

TL;DR

This paper proves the existence of large sets of polynomials with integer coefficients that are dynamically distinguishable mod p for most primes, using Galois theory and wreath product statistics.

Contribution

It introduces a method to construct infinitely many polynomial sets that are dynamically distinguishable mod p, generalizing Morton's Galois group results.

Findings

Existence of infinitely many dynamically distinguishable polynomial sets

Use of Galois theory and wreath product analysis

Generalization of Morton's work on dynatomic polynomial Galois groups

Abstract

A polynomial with integer coefficients yields a family of dynamical systems indexed by primes as follows: for any prime $p$, reduce its coefficients mod $p$ and consider its action on the field $\mathbb{F}_p$. We say a subset of $\mathbb{Z}[x]$ is dynamically distinguishable mod $p$ if the associated mod $p$ dynamical systems are pairwise non-isomorphic. For any $k,M\in\mathbb{Z}_{>1}$, we prove that there are infinitely many sets of integers $\mathcal{M}$ of size $M$ such that $\left\{ x^k+m\mid m\in\mathcal{M}\right\}$ is dynamically distinguishable mod $p$ for most $p$ (in the sense of natural density). Our proof uses the Galois theory of dynatomic polynomials largely developed by Morton, who proved that the Galois groups of these polynomials are often isomorphic to a particular family of wreath products. In the course of proving our result, we generalize Morton's work and compute statistics of these wreath products.