A converse of dynamical Mordell--Lang conjecture in positive characteristic
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909455273164800 |
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| author | Lee, Jungin Nam, Gyeonghyeon |
| author_facet | Lee, Jungin Nam, Gyeonghyeon |
| contents | In this paper, we prove the converse of the dynamical Mordell--Lang conjecture in positive characteristic: For every subset $S \subseteq \mathbb{N}_0$ which is a union of finitely many arithmetic progressions along with finitely many $p$-sets of the form $\left \{ \sum_{j=1}^{m} c_j p^{k_jn_j} : n_j \in \mathbb{N}_0 \right \}$ ($c_j \in \mathbb{Q}$, $k_j \in \mathbb{N}_0$), there exist a split torus $X = \mathbb{G}_m^k$ defined over $K=\overline{\mathbb{F}_p}(t)$, an endomorphism $Φ$ of $X$, $α\in X(K)$ and a closed subvariety $V \subseteq X$ such that $\left \{ n \in \mathbb{N}_0 : Φ^n(α) \in V(K) \right \} = S$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_05107 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A converse of dynamical Mordell--Lang conjecture in positive characteristic Lee, Jungin Nam, Gyeonghyeon Number Theory Dynamical Systems In this paper, we prove the converse of the dynamical Mordell--Lang conjecture in positive characteristic: For every subset $S \subseteq \mathbb{N}_0$ which is a union of finitely many arithmetic progressions along with finitely many $p$-sets of the form $\left \{ \sum_{j=1}^{m} c_j p^{k_jn_j} : n_j \in \mathbb{N}_0 \right \}$ ($c_j \in \mathbb{Q}$, $k_j \in \mathbb{N}_0$), there exist a split torus $X = \mathbb{G}_m^k$ defined over $K=\overline{\mathbb{F}_p}(t)$, an endomorphism $Φ$ of $X$, $α\in X(K)$ and a closed subvariety $V \subseteq X$ such that $\left \{ n \in \mathbb{N}_0 : Φ^n(α) \in V(K) \right \} = S$. |
| title | A converse of dynamical Mordell--Lang conjecture in positive characteristic |
| topic | Number Theory Dynamical Systems |
| url | https://arxiv.org/abs/2403.05107 |