Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On higher dimensional integrality and multiplicative dependence in semigroup algebraic dynamics

Jorge Mello, Yu Yasufuku

Abstract

We study multiplicative dependence of points in semigroup orbits in higher dimensions. More specifically, we show that the non-density of integral points in semigroup orbits implies sparsity of multiplicative dependence in orbits. This can be viewed as a semigroup dynamical and a higher dimensional version of recent results by Bérczes, Ostafe, Shparlinski and Silverman, which in turn can be viewed as a generalization of theorems of Northcott and Siegel. We also confirm that the non-density hypothesis of integral points in orbits is implied by Vojta's conjecture.

On higher dimensional integrality and multiplicative dependence in semigroup algebraic dynamics

Abstract

We study multiplicative dependence of points in semigroup orbits in higher dimensions. More specifically, we show that the non-density of integral points in semigroup orbits implies sparsity of multiplicative dependence in orbits. This can be viewed as a semigroup dynamical and a higher dimensional version of recent results by Bérczes, Ostafe, Shparlinski and Silverman, which in turn can be viewed as a generalization of theorems of Northcott and Siegel. We also confirm that the non-density hypothesis of integral points in orbits is implied by Vojta's conjecture.

Paper Structure

This paper contains 4 sections, 4 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on heights
  3. Proofs of the main theorems
  4. Regarding Hypothesis \ref{['eq:thesetfggp']} and Vojta's conjecture

Key Result

Theorem 1.1

Let $k$ be a number field, and $\Gamma$ be a finitely generated subgroup of $\mathbb{G}_m^N(k) \subset \mathbb{P}^N(k)$. For $i=1, \ldots, \ell$, let $\phi_i$ be an endomorphism of $\mathbb P^N$ of degree $d_i \ge 2$ defined over a number field $k$, and let $\mathcal{F}$ be the semi-group generated is contained in $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • proof : Proof of Theorem \ref{['thm:fggp']}
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • proof : Proof of Theorem \ref{['thm:fggp2']}
  • Remark 3.4
  • Remark 3.5
  • ...and 6 more