有理函数域上单临界多项式的预周期图
Preperiodic portraits for unicritical polynomials over a rational function field
摘要：
设$K$是特征为零的代数闭域，$\mathcal{K} := K(t)$是$K$上的有理函数域.对于每个$d \ge 2$，我们考虑单临界多项式$f_d(z) := z^d + t \in \mathcal{K}[z]$，我们问以下问题：如果我们固定$α\in \mathcal{K}$和整数$M \ge 0$，$N \ge 1$和$d \ge 2$，是否存在一个地方$\mathfrak{p} \in \operatorname{Spec} K[t]$，使得，模$\mathfrak{p}$，点$α$进入$N$ -循环后，正是$M$步骤下迭代$f_d$？我们完全回答这个问题，得出的结论是，答案一般是肯定的，并明确给出所有的反例。这扩展了作者以前的工作，在$α$是一个恒定点的情况下。
英文摘要：
Let $K$ be an algebraically closed field of characteristic zero, and let $\mathcal{K} := K(t)$ be the rational function field over $K$. For each $d \ge 2$, we consider the unicritical polynomial $f_d(z) := z^d + t \in \mathcal{K}[z]$, and we ask the following question: If we fix $α\in \mathcal{K}$ and integers $M \ge 0$, $N \ge 1$, and $d \ge 2$, does there exist a place $\mathfrak{p} \in \operatorname{Spec} K[t]$ such that, modulo $\mathfrak{p}$, the point $α$ enters into an $N$-cycle after precisely $M$ steps under iteration by $f_d$? We answer this question completely, concluding that the answer is generally affirmative and explicitly giving all counterexamples. This extends previous work by the author in the case that $α$ is a constant point.
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