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

R. Padhy, S. S. Rout

TL;DR

The paper addresses the problem of bounding the number of $S$-integral points in backward orbits under the power map $oldsymbol{igl ext{varphi}igr}(z)=z^d$ for a non-preperiodic point $oldsymbol{eta}$ in a number field $K$. It combines canonical heights, $S$-integrality via local chordal metrics, Berkovich dynamics, and the Arakelov-Zhang pairing with quantitative equidistribution and linear forms in logarithms to obtain uniform control over the Galois orbits of backward-iterate points that are $S$-integral relative to $oldsymbol{eta}$. The main results give explicit uniform bounds on the size of Galois orbits, first for a fixed field (Theorem 2) and then uniformly over extensions of bounded degree (Theorem 02), with the latter leveraging height lower bounds (Dobrowolski) and a finiteness argument. The work advances uniformity in unlikely-intersections-type questions in arithmetic dynamics and provides a framework combining dynamical and Diophantine tools to constrain backward-orbit integrality, offering effective consequences for backward-iterates of monomial maps.

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 $β$ in $K$, relative to a non-preperiodic point $α\in \mathbb{P}^1(\overline{K})$, under the power map $\varphi(z) =z^d $.

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

TL;DR

The paper addresses the problem of bounding the number of -integral points in backward orbits under the power map for a non-preperiodic point in a number field . It combines canonical heights, -integrality via local chordal metrics, Berkovich dynamics, and the Arakelov-Zhang pairing with quantitative equidistribution and linear forms in logarithms to obtain uniform control over the Galois orbits of backward-iterate points that are -integral relative to . The main results give explicit uniform bounds on the size of Galois orbits, first for a fixed field (Theorem 2) and then uniformly over extensions of bounded degree (Theorem 02), with the latter leveraging height lower bounds (Dobrowolski) and a finiteness argument. The work advances uniformity in unlikely-intersections-type questions in arithmetic dynamics and provides a framework combining dynamical and Diophantine tools to constrain backward-orbit integrality, offering effective consequences for backward-iterates of monomial maps.

Abstract

Let be a number field with algebraic closure and let be a finite set of places of containing all the archimedean places. It is known from Silverman's result that a forward orbit of a rational map contains finitely many -integers in the number field K when is not a polynomial. Sookdeo stated an analogous conjecture for the backward orbits of a rational map using a general -integrality notion based on the Galois conjugates of points. He proved his conjecture for the power map for and consequently for Chebyshev maps (J. Number Theory 131 (2011), 1229-1239). In this paper, we establish uniform bounds on the number of -integral points in the backward orbits of any non-zero in , relative to a non-preperiodic point , under the power map .
Paper Structure (14 sections, 14 theorems, 125 equations)

This paper contains 14 sections, 14 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Heights
  4. $S$-integrality
  5. The Berkovich affine and projective lines
  6. The Arakelov-Zhang pairing
  7. Quantitative equidistribution
  8. Linear forms in logarithms
  9. Quantitative logarithmic equidistribution
  10. Finiteness of S-integral points
  11. Proof of Proposition \ref{['prop01']}
  12. Proof of Main Results
  13. Proof of Theorem \ref{['thm2']}
  14. Proof of Theorem \ref{['thm02']}

Key Result

Theorem 1.2

Let $K$ be a number field and $S$ be a finite set of places of $K$ including all the archimedean places. Let $d\geq 2$ be an integer, $\varphi(z)=z^d$ be a rational map and let $\alpha \not \in \emph{PrePer}(\varphi, \overline{K})$. Then there exists a constant $C=C([K:\mathbb{Q}],|S|, \varphi)$ su

Theorems & Definitions (24)

  • Conjecture 1.1: Sookdeo, p.1230
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1: petsche2012, Corollary 12
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 14 more