Uniform bounds on $S$-integral points in backward orbits

Let $K$ be a number field with algebraic closure $\overline{K}$ and let $S$ be a finite set of places of $K$ containing all the archimedean places. It is known from Silverman's result that a forward orbit of a rational map $\varphi$ contains finitely many $S$-integers in the number field K when $\varphi^2$ is not a polynomial. Sookdeo stated an analogous conjecture for the backward orbits of a rational map $\varphi$ using a general $S$-integrality notion based on the Galois conjugates of points. He proved his conjecture for the power map $\varphi(z) =z^d$ for $d \geq 2$ and consequently for Chebyshev maps (J. Number Theory 131 (2011), 1229-1239). In this paper, we establish uniform bounds on the number of $S$-integral points in the backward orbits of any non-zero $\beta$ in $K$, relative to a non-preperiodic point $\alpha \in \mathbb{P}^1(\overline{K})$, under the power map $\varphi(z) =z^d$.