标题： 原点在单临界多项式映射下的有理迭代前像的数量 显示英文标题

标题： The number of rational iterated preimages of the origin under unicritical polynomial maps

摘要： 我们研究在单临界映射$f_{d,c}(x)=x^d+c$下原点的有理迭代前像。Faber--Hutz--Stoll 和 Hutz--Hyde--Krause 之前的工作在二次情况下建立了有限性和条件性界限。在此基础上，我们证明对于$d=2$和$c \in \mathbb Q\setminus\{0,-1\}$，原点没有有理四阶前像，而对于所有$d \geq 3$，在平凡情况之外原点没有有理二阶前像。证明依赖于前像曲线的几何分析、椭圆 Chabauty 方法和丢番图约简。作为结果，我们确定了在$f_{d,c}$下$0$的有理迭代前像的数量，对于所有$d\geq 2$。 显示英文摘要

摘要： We study rational iterated preimages of the origin under unicritical maps $f_{d,c}(x)=x^d+c$. Earlier works of Faber--Hutz--Stoll and Hutz--Hyde--Krause established finiteness and conditional bounds in the quadratic case. Building on this, we prove that for $d=2$ and $c \in \mathbb Q\setminus\{0,-1\}$ there are no rational fourth preimages of the origin, and for all $d \geq 3$ there are no rational second preimages outside trivial cases. The proof relies on geometric analysis of preimage curves, the elliptic Chabauty method, and Diophantine reduction. As a result, we determine the number of rational iterated preimages of $0$ under $f_{d,c}$ for all $d\geq 2$.