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 $φ$ contains finitely many $S$-integers in the number field K when $φ^2$ is not a polynomial. Sookdeo stated an analogous conjecture for the backward orbits of a rational map $φ$ using a general $S$-integrality notion based on the Galois conjugates of points. He proved his conjecture for the power map $φ(z) =z^d$ 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 $β$ in $K$, relative to a non-preperiodic point $α\in \mathbb{P}^1(\overline{K})$, under the power map $φ(z) =z^d $.

💡 Research Summary

The paper investigates the distribution of S‑integral points in backward orbits of the power map ϕ(z)=z^d (d≥2) over a number field K. Building on Silverman’s finiteness theorem for forward orbits and Sookdeo’s conjecture for backward orbits, the authors provide explicit uniform bounds on the size of Galois orbits of such integral points.

The main objects are: a finite set S of places of K containing all archimedean places, a non‑preperiodic point α∈ℙ^1(K), and a non‑zero β∈K. A point γ∈ℙ^1(K) is called S‑integral relative to α if, for every place v∉S and every pair of Galois embeddings σ,τ, the local chordal distance λ_{α,v}(σ(γ)) vanishes; equivalently, the v‑adic distances between the Galois conjugates of γ and those of α satisfy the usual integrality inequalities.

The backward orbit of β under ϕ is O⁻ϕ(β)=⋃{n≥0}{γ∈K | γ^{d^n}=β}. When β is neither zero nor a root of unity, each γ is a d^n‑th root of β, and its Galois orbit size |G_K(γ)| is essentially d^n. The authors show that the S‑integrality condition forces n to be bounded, which in turn yields a uniform bound on |G_K(γ)|.

Theorem 1.2 asserts the existence of a constant C=C(