Dynamically distinguishing polynomials
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 nonisomorphic. 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2016
 arXiv:
 arXiv:1609.09186
 Bibcode:
 2016arXiv160909186B
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematics  Number Theory;
 37P05 (Primary);
 37P25;
 11R32;
 20B35 (Secondary)
 EPrint:
 18 pages, updated bibliography, updated acknowledgements, and modified notation