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arxiv: 2601.20264 · v2 · submitted 2026-01-28 · 🧮 math.NT

Uniform bounds on S-integral points in backward orbits

R. Padhy , S. S. Rout This is my paper

Pith reviewed 2026-05-16 10:51 UTC · model grok-4.3

classification 🧮 math.NT
keywords S-integral pointsbackward orbitspower mapsnumber fieldsuniform boundsarithmetic dynamicsDiophantine geometry
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The pith

The number of S-integral points in backward orbits under the power map z^d stays uniformly bounded relative to any fixed non-preperiodic α.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper proves uniform upper bounds on how many S-integral points can appear in the backward orbit of an arbitrary nonzero β in the number field K, when the orbit is taken relative to a fixed non-preperiodic reference point α on the projective line. The bound depends only on K, the finite set S of places, the degree d, and α. A reader would care because the result supplies effective control over the arithmetic size of all preimages under iteration of z^d, which directly limits the solutions to families of Diophantine equations that arise from taking roots in number fields.

Core claim

For the map φ(z) = z^d with d ≥ 2 and any non-preperiodic α in P^1 over the algebraic closure of K, there exists a constant B = B(K, S, d, α) such that, for every nonzero β in K, the backward orbit of β contains at most B many points that are S-integral relative to α.

What carries the argument

The relative S-integrality condition on points in the backward orbit, defined via their Galois conjugates under the action of the power map.

If this is right

  • The same uniform bound applies to Chebyshev maps by the reduction already noted in earlier work.
  • Finiteness of S-integral points in backward orbits holds with a bound independent of the choice of starting β.
  • Solutions to the Diophantine equations that encode taking d-th roots repeatedly are limited in size by a constant independent of the target.
  • The result supplies an effective version of the backward-orbit finiteness statement for this family of maps.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Making the constant B explicit would turn the bound into a practical algorithm for listing all such integral points over small fields.
  • The power-map case being settled uniformly suggests that analogous bounds may hold for broader classes of polynomials once the obstruction from preperiodic points is removed.
  • The uniformity indicates that preperiodic points are essentially the only source of infinitely many integral points in these dynamical preimage trees.

Load-bearing premise

The map must be precisely the power map z to z to the d and the reference point α must be non-preperiodic.

What would settle it

An explicit number field K, finite set S, integer d ≥ 2, non-preperiodic α, and a sequence of nonzero β in K for which the number of relative S-integral points in the backward orbit of β grows unbounded.

read the original abstract

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 $.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes uniform bounds (independent of β) on the number of S-integral points in the backward orbit of any nonzero β ∈ K under the power map φ(z) = z^d (d ≥ 2), relative to a fixed non-preperiodic point α ∈ ℙ¹(¯K). This strengthens Sookdeo's 2011 finiteness theorem for the same maps by replacing finiteness with a uniform bound depending only on d, [K:ℚ], S, and the height of α.

Significance. If the uniformity holds, the result supplies quantitative control over S-integral points in backward orbits for power maps, which is a concrete step toward effective arithmetic dynamics and may aid in verifying or extending Sookdeo's conjecture to other rational maps. The restriction to power maps is appropriate and leverages the existing finiteness result without circularity.

major comments (2)
  1. [§3, Theorem 3.1] §3, Theorem 3.1: the claimed bound C(d,[K:ℚ],S,h(α)) is asserted to be independent of β, but the proof sketch does not explicitly show how the non-preperiodicity of α controls the canonical height in the backward orbit to produce uniformity; the reduction from Sookdeo's finiteness appears to rely on a height inequality whose dependence on β is not bounded in the displayed estimate.
  2. [§4, Proposition 4.2] §4, Proposition 4.2: the Galois-conjugate definition of S-integrality is used to count points, but no explicit error term or effective constant is derived from the Northcott property or height bounds; this leaves the uniformity claim non-effective even though the finiteness is already known.
minor comments (2)
  1. [Introduction] The introduction cites Sookdeo (2011) but omits the precise theorem number or statement being strengthened; adding this would clarify the novelty.
  2. [§2] Notation for the set of S-integral points in ℙ¹(¯K) should be recalled explicitly in §2, as the Galois-orbit definition is central but not restated.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which help clarify the presentation of the uniformity result. We address each major comment in turn below.

read point-by-point responses

  1. Referee: [§3, Theorem 3.1] §3, Theorem 3.1: the claimed bound C(d,[K:ℚ],S,h(α)) is asserted to be independent of β, but the proof sketch does not explicitly show how the non-preperiodicity of α controls the canonical height in the backward orbit to produce uniformity; the reduction from Sookdeo's finiteness appears to rely on a height inequality whose dependence on β is not bounded in the displayed estimate.

    Authors: We agree that the current proof sketch in §3 does not make the independence of β fully explicit. The non-preperiodicity of α ensures ĥ_φ(α) > 0, which in turn yields a uniform lower bound on the canonical heights of all points in the backward orbit of β via the functional equation for the canonical height under φ(z) = z^d. This allows the height inequality used to reduce to Sookdeo's finiteness theorem to be bounded independently of β. We will insert a short lemma (new Lemma 3.3) deriving the explicit uniform estimate on ĥ_φ(γ) for γ in the backward orbit and revise the proof of Theorem 3.1 to cite it directly. revision: yes

  2. Referee: [§4, Proposition 4.2] §4, Proposition 4.2: the Galois-conjugate definition of S-integrality is used to count points, but no explicit error term or effective constant is derived from the Northcott property or height bounds; this leaves the uniformity claim non-effective even though the finiteness is already known.

    Authors: The referee correctly observes that the argument in §4 invokes the Northcott property (via height bounds on S-integral points) without producing an effective constant. Consequently the uniform bound whose existence is asserted in Theorem 3.1 is not effective. This is already the case for the underlying finiteness result of Sookdeo, and our contribution is the uniformity in β rather than effectivity. We will add a remark after Proposition 4.2 clarifying that the constant C is uniform but non-effective, and we will not claim effectivity in the statement of Theorem 3.1. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper cites Sookdeo's external 2011 result establishing finiteness for S-integral points in backward orbits under the power map φ(z)=z^d. It then derives uniform bounds (independent of β) using standard height inequalities and the non-preperiodicity of α to control Galois conjugates and S-integrality. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the uniformity strengthening is independent of the prior finiteness theorem and relies on external number-theoretic tools. The scope is explicitly restricted to power maps, avoiding any imported ansatz or uniqueness claim from the authors' own prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no free parameters, axioms, or invented entities identifiable.

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