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A SPARSITY RESULT FOR THE DYNAMICAL MORDELL–LANG CONJECTURE IN POSITIVE CHARACTERISTIC

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Abstract We prove a quantitative partial result in support of the dynamical Mordell–Lang conjecture (also known as the DML conjecture) in positive characteristic. More precisely, we show the following: given… Click to show full abstract

Abstract We prove a quantitative partial result in support of the dynamical Mordell–Lang conjecture (also known as the DML conjecture) in positive characteristic. More precisely, we show the following: given a field K of characteristic p, a semiabelian variety X defined over a finite subfield of K and endowed with a regular self-map $\Phi :X{\longrightarrow } X$ defined over K, a point $\alpha \in X(K)$ and a subvariety $V\subseteq X$ , then the set of all nonnegative integers n such that $\Phi ^n(\alpha )\in V(K)$ is a union of finitely many arithmetic progressions along with a subset S with the property that there exists a positive real number A (depending only on X, $\Phi $ , $\alpha $ and V) such that for each positive integer M, $$\begin{align*}\scriptsize\#\{n\in S\colon n\le M\}\le A\cdot (1+\log M)^{\dim V}.\end{align*}$$

Keywords: lang conjecture; mordell lang; dynamical mordell; conjecture positive; positive characteristic; conjecture

Journal Title: Bulletin of the Australian Mathematical Society
Year Published: 2021

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