新的可约多项式迭代
Newly reducible polynomial iterates
摘要：
给定一个域$K$和$n > 1$，如果$f^{n-1}$在$K$上不可约，但$f^n$不是，则我们说多项式$f \in K[x]$在$K$上有新的可约$n$ th_n，（这里$f^i$表示$f$的$i$ th_n）。我们提出的问题的特征，对于给定的$d,n > 1$，领域$K$，使存在$f \in K[x]$度$d$与新的可约$n$ th，和类似的问题，领域承认无限多这样的$f$。我们给出结果的情况下，$(d,n) \in \{(2,2), (2,3), (3,2), (4,2)\}$以及$(d,2)$时，$d \equiv 2 \bmod{4}$。特别是，我们表明，对于所有这些$(d,n)$对，有无限多幺半群$f \in \mathbb{Z}[x]$的程度$d$与新的可约$n$第四遍$\mathbb{Q}$。奇怪的是，黄金分割率的最小多项式$x^2 - x - 1$是$f \in \mathbb{Z}[x]$的一个例子，它具有新的可约的第三个系数;很少有其他例子具有小的系数。我们的调查引发了一些质疑和未决问题。
英文摘要：
Given a field $K$ and $n > 1$, we say that a polynomial $f \in K[x]$ has newly reducible $n$th iterate over $K$ if $f^{n-1}$ is irreducible over $K$, but $f^n$ is not (here $f^i$ denotes the $i$th iterate of $f$). We pose the problem of characterizing, for given $d,n > 1$, fields $K$ such that there exists $f \in K[x]$ of degree $d$ with newly reducible $n$th iterate, and the similar problem for fields admitting infinitely many such $f$. We give results in the cases $(d,n) \in \{(2,2), (2,3), (3,2), (4,2)\}$ as well as for $(d,2)$ when $d \equiv 2 \bmod{4}$. In particular, we show that for all these $(d,n)$ pairs, there are infinitely many monic $f \in \mathbb{Z}[x]$ of degree $d$ with newly reducible $n$th iterate over $\mathbb{Q}$. Curiously, the minimal polynomial $x^2 - x - 1$ of the golden ratio is one example of $f \in \mathbb{Z}[x]$ with newly reducible third iterate; very few other examples have small coefficients. Our investigations prompt a number of conjectures and open questions.
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