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Dynamical GCD Problems and a Variant of the Dynamical Mordell-Lang Conjecture

She Yang, Xiao Zhong

Abstract

In \cite{NZ25}, the authors resolved the rational function analogue of the finiteness results for greatest common divisors of iterates of polynomials established in \cite{HT17}. These results may be viewed as dynamical generalizations of a classical problem concerning upper bounds for the greatest common divisors (GCDs) of two integer sequences studied by Bugeaud, Corvaja, and Zannier. The most delicate case arises when the maps involved are automorphisms, where the methods of \cite{NZ25} and \cite{HT17} rely heavily on Diophantine approximation and asymptotic analysis. In the present paper, we develop an alternative approach to the automorphism case. This method is more powerful, allowing us to give complete answers to the further questions posed in \cite{HT17}. In particular, we strengthen the main theorem of \cite{HT17} and provide an alternative proof of the main theorem of \cite{NZ25} in the automorphism setting. Moreover, we relate this dynamical GCD problem to a special case of a higher-dimensional generalization of the Dynamical Mordell--Lang Conjecture proposed by Junyi Xie. We establish this generalized conjecture when the dynamics arise from algebraic group actions. In addition, we resolve the corresponding special case associated with dynamical GCD questions when the maps involved are polynomials.

Dynamical GCD Problems and a Variant of the Dynamical Mordell-Lang Conjecture

Abstract

In \cite{NZ25}, the authors resolved the rational function analogue of the finiteness results for greatest common divisors of iterates of polynomials established in \cite{HT17}. These results may be viewed as dynamical generalizations of a classical problem concerning upper bounds for the greatest common divisors (GCDs) of two integer sequences studied by Bugeaud, Corvaja, and Zannier. The most delicate case arises when the maps involved are automorphisms, where the methods of \cite{NZ25} and \cite{HT17} rely heavily on Diophantine approximation and asymptotic analysis. In the present paper, we develop an alternative approach to the automorphism case. This method is more powerful, allowing us to give complete answers to the further questions posed in \cite{HT17}. In particular, we strengthen the main theorem of \cite{HT17} and provide an alternative proof of the main theorem of \cite{NZ25} in the automorphism setting. Moreover, we relate this dynamical GCD problem to a special case of a higher-dimensional generalization of the Dynamical Mordell--Lang Conjecture proposed by Junyi Xie. We establish this generalized conjecture when the dynamics arise from algebraic group actions. In addition, we resolve the corresponding special case associated with dynamical GCD questions when the maps involved are polynomials.
Paper Structure (14 sections, 31 theorems, 405 equations)

This paper contains 14 sections, 31 theorems, 405 equations.

Table of Contents

  1. Introduction
  2. Historical Background
  3. A New Approach to the Automorphisms Cases
  4. Further applications in characteristic $0$
  5. Connection to the Dynamical Mordell--Lang Conjecture
  6. Outline of the paper
  7. Common Zeros of Iterated Möbius Transformations
  8. On Question \ref{['qu: automorphisms-common-zeros']}
  9. Applications on the system of $\{f,g,c\}$
  10. Positive Characteristic Case
  11. Connection to the Generalized Dynamical Mordell-Lang Problem
  12. Special Cases Related to Common Zeros of Iterated Morphisms
  13. Preliminaries on elementary number theory
  14. Special case of Question \ref{['qu: DML-varieties']} when $f$, $g$ and $c$ are polynomials

Key Result

Theorem 1.4

Let $f(x) = \alpha x + \beta$ and $g(x) = \delta x + \gamma$, where $\alpha, \delta \in {\mathbb C}^*$ and $\beta, \gamma \in {\mathbb C}$. Let $c_1$ and $c_2$ be two rational functions such that $c_1 \not\in \{f^n:n \in {\mathbb N}^+\}$ and $c_2 \not\in \{g^n:n\in{\mathbb N}^+\}$. Then there are in holds for some $m,n \in {\mathbb N}^+$ implies the following holds:

Theorems & Definitions (69)

  • Definition 1.1
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Theorem 1.9
  • Theorem 1.10
  • Example 1.12
  • Theorem 1.13
  • Theorem 1.14
  • ...and 59 more