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

Abstract

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 $d\geq 2$.