Dynamically distinguishing polynomials Article Swipe
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· 2017
· Open Access
·
· DOI: https://doi.org/10.1186/s40687-017-0103-3
· OA: W2950419884
A polynomial with integer coefficients yields a family of dynamical systems\nindexed by primes as follows: for any prime $p$, reduce its coefficients mod\n$p$ and consider its action on the field $\\mathbb{F}_p$. We say a subset of\n$\\mathbb{Z}[x]$ is dynamically distinguishable mod $p$ if the associated mod\n$p$ dynamical systems are pairwise non-isomorphic. For any\n$k,M\\in\\mathbb{Z}_{>1}$, we prove that there are infinitely many sets of\nintegers $\\mathcal{M}$ of size $M$ such that $\\left\\{ x^k+m\\mid\nm\\in\\mathcal{M}\\right\\}$ is dynamically distinguishable mod $p$ for most $p$\n(in the sense of natural density). Our proof uses the Galois theory of\ndynatomic polynomials largely developed by Morton, who proved that the Galois\ngroups of these polynomials are often isomorphic to a particular family of\nwreath products. In the course of proving our result, we generalize Morton's\nwork and compute statistics of these wreath products.\n
