论文标题
新还原多项式迭代
Newly reducible polynomial iterates
论文作者
论文摘要
给定一个字段$ k $和$ n> 1 $，我们说k [x] $中的多项式$ f \如果$ f^{n-1} $在$ k $上不可记述，$ f^n $不可记录，但是$ f^n $ note $ f^n $不可记录（在这里$ f^i $ f^i $ deotes $ f^i $ i $ f $ f $ f $ f $）。我们提出了表征的问题，对于给定的$ d，n> 1 $，fields $ k $，因此存在$ f \ in k [x] $ d $ d $的$ f \ with $ d $，而新还原$ n $ th itterate，以及承认无限许多此类$ f $的字段的类似问题。我们在情况下给出了$（d，n）\ in \ in \ {（2,2），（2,3），（3,2），（4,2）\} $的结果，以及$（d，2）$当$ d \ equiv 2 \ equiv 2 \ bmod {4} $。特别是，我们表明，对于所有这些$（d，n）$ pairs，有无限的许多monic $ f \ in \ mathbb {z} [x] $ $ d $的$ a $ d $，新还原$ n $ th $ n $ th iTerate超过$ \ mathbb {q} $。奇怪的是，黄金比率的最小多项式$ x^2 -x -1 $是\ Mathbb {z} $ in \ mathbb {z} [x] $的一个示例，新还原的第三个迭代;其他示例很少有系数。我们的调查促使许多猜想和开放问题。
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.
