动态可区别多项式
Dynamically distinguishing polynomials
摘要：
一个整数系数的多项式产生一个由素数索引的动力系统族如下：对于任何素数$p$，减少其系数mod $p$，并考虑其对域$\mathbb{F}_p$的作用。我们说$\mathbb{Z}[x]$的子集是动态可区分的mod $p$，如果相关的mod $p$动力系统是成对非同构的。对于任何$k,M\in\mathbb{Z}_{>1}$，我们证明，有无穷多组整数$\mathcal{M}$的大小$M$，使$\left\{ x^k+m\mid m\in\mathcal{M}\right\}$是动态可区分的mod $p$的大多数$p$（在自然密度的意义上）。我们的证明使用伽罗瓦理论的dynatomic多项式主要是由莫顿，谁证明了伽罗瓦集团的这些多项式往往同构于一个特定的家庭的圈积。在证明结果的过程中，我们推广了Morton的工作，并计算了这些圈积的统计量.
英文摘要：
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.
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