A sparsity result for the Dynamical Mordell-Lang Conjecture in positive characteristic
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...
| Main Authors: | , , , |
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| Format: | Article in Journal/Newspaper |
| Language: | unknown |
| Published: |
arXiv
2020
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| Subjects: | |
| Online Access: | https://dx.doi.org/10.48550/arxiv.2012.13711 https://arxiv.org/abs/2012.13711 |
| Summary: | 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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