Mathematical Research Letters

Volume 20 (2013)

Number 3

On a uniform bound for the number of exceptional linear subvarieties in the dynamical Mordell-Lang conjecture

Pages: 547 – 566

DOI: https://dx.doi.org/10.4310/MRL.2013.v20.n3.a12

Authors

Joseph H. Silverman (Department of Mathematics, Brown University, Providence, Rhode Island, U.S.A.)

Bianca Viray (Department of Mathematics, Brown University, Providence, Rhode Island, U.S.A.)

Abstract

Let $\phi : \mathbb{P}^n \to \mathbb{P}^n$ be a morphism of degree $d \geq 2$ defined over $\mathbb{C}$. The dynamical Mordell–Lang conjecture says that the intersection of an orbit $\mathcal{O}_{\phi} (P)$ and a subvariety $X \subset \mathbb{P}^n$ is usually finite.We consider the number of linear subvarieties $L \subset \mathbb{P}^n$ such that the intersection $O_{\phi} (P)\cap L$ is “larger than expected.” When $\phi $ is the $d^{th}$-power map and the coordinates of $P$ are multiplicatively independent, we prove that there are only finitely many linear subvarieties that are “super-spanned” by $O_{\phi} (P)$, and further that the number of such subvarieties is bounded by a function of $n$, independent of the point $P$ and the degree $d$. More generally, we show that there exists a finite subset $S$, whose cardinality is bounded in terms of $n$, such that any $n + 1$ points in $O_{\phi} (P) \setminus S$ are in linear general position in $\mathbb{P}n$.

Keywords

arithmetic dynamics, Mordell-Lang conjecture

2010 Mathematics Subject Classification

Primary 37P15. Secondary 11D45, 37P05.

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Published 9 January 2014