#### Published In

Research in the Mathematical Sciences

#### Document Type

Article

#### Publication Date

7-10-2017

#### Subjects

Finite fields (Algebra), Galois theory, Wreath products (Group theory), 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 FpFp. We say a subset of Z[x]Z[x] is dynamically distinguishable mod p if the associated mod pdynamical systems are pairwise non-isomorphic. For any k,M∈Z>1k,M∈Z>1, we prove that there are infinitely many sets of integers MM of size M such that {xk+m∣m∈M}{xk+m∣m∈M} 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.

#### DOI

10.1186/s40687-017-0103-3

#### Persistent Identifier

http://pdxscholar.library.pdx.edu/mth_fac/197/

#### Citation Details

Bridy, A., & Garton, D. (2017). Dynamically distinguishing polynomials. Research in the Mathematical Sciences, 4(1), 13.

## Description

© The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.