标题： 在高次数下迭代的$x^d+c$的分解 显示英文标题

标题： On the factorization of iterates of $x^d+c$ in large degree

摘要： 设$K$为特征零的曲线的函数域或一个数域，在该数域上$abc$猜想成立，固定$\alpha\in K$，并让$f_{d,c}(x)=x^d+c$对于某些$d\geq2$和某些$c\in K$。 然后对于许多$c$和$d$，我们证明对于所有$n\geq1$，$f_{d,c}^n(x)-\alpha$在$K[x]$中最多有$d$个因子。 例如，当 $\alpha=0$ 我们证明集合 \[\Big\{d\,:\, f_{d,c}^n(x)\;\text{has at most $d$ factors in $K[x]$ for all $n\geq1$ and all $h(c)>0$}\Big\}\] 具有正的渐近密度。 我们随后将这一结果应用于计算某些正向轨道中素数因子的密度，并建立某些逆向轨道中整数点的有限性。 显示英文摘要

摘要： Let $K$ be a function field of a curve in characteristic zero or a number field over which the $abc$-conjecture holds, fix $\alpha\in K$, and let $f_{d,c}(x)=x^d+c$ for some $d\geq2$ and some $c\in K$. Then for many $c$ and $d$, we prove that $f_{d,c}^n(x)-\alpha$ has at most $d$ factors in $K[x]$ for all $n\geq1$. For example, when $\alpha=0$ we prove that the set \[\Big\{d\,:\, f_{d,c}^n(x)\;\text{has at most $d$ factors in $K[x]$ for all $n\geq1$ and all $h(c)>0$}\Big\}\] has positive asymptotic density. We then apply this result to compute the density of prime divisors in certain forward orbits and to establish the finiteness of integral points in certain backward orbits.