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arXiv 2012.13711math.NT
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正特征动力学Mordell-Lang猜想的一个稀疏性结果

A sparsity result for the Dynamical Mordell-Lang Conjecture in positive characteristic

Dragos Ghioca, Alina Ostafe, Sina Saleh, Igor E. Shparlinski

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摘要：

本文证明了一个定量的部分结果，支持动力学Mordell-Lang猜想（也称为DML猜想）的正特征。更准确地说，我们展示了以下内容：给定特征$p$的域$K$，给定在$K$的有限子域上定义的并被赋予在$K$上定义的正则自映射$Φ:X \longrightarrow X$的半阿贝尔簇$X$，给定点$α\in X(K)$和子簇$V\subseteq X$，则所有非-负整数$n$，使得$Φ^n(α)\in V(K)$是多个算术级数连同子集$S$的并集，具有以下性质：存在正实数$A$（仅取决于$N$、$Φ$、$α$、$V$），使得对于每个正整数$M$，我们有$$\#\left\{n\in S\colon~ n\le M\right\}\le A\cdot \left(1+\log M\right)^{\dim V}.$$

英文摘要：

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 $p$, given a semiabelian variety $X$ defined over a finite subfield of $K$ and endowed with a regular self-map $Φ:X \longrightarrow X$ defined over $K$, given a point $α\in X(K)$ and a subvariety $V\subseteq X$, then the set of all non-negative integers $n$ such that $Φ^n(α)\in V(K)$ 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 $N$, $Φ$, $α$, $V$) such that for each positive integer $M$, we have $$\#\left\{n\in S\colon~ n\le M\right\}\le A\cdot \left(1+\log M\right)^{\dim V}.$$

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