关于正向轨道上 (D,S) 积分点的非扎里斯基密度和子空间定理

Pub Date : 2024-07-17 DOI:10.1016/j.jnt.2024.06.005
Nathan Grieve , Chatchai Noytaptim
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引用次数: 0

摘要

在基数域上,我们研究了有理点的前-轨道上-积分点的扎里斯基非密度这一有吸引力的问题。这里, 是几何上不可还原的投影变种在 .给定一个在 和 上定义的非零且有效的准极化卡蒂埃除数,我们的主要结果给出了一个充分条件,这个充分条件是根据 、 的动态定义的某些动态子集的非扎里斯基密度来表述的。 对于积分点的情况,这个结果给出了在 的积分点的非扎里斯基密度的充分条件。我们的方法是在 Yasufuku 的基础上发展而来的,是建立在 Silverman 早期工作的基础上的。我们的主要结果给出了施密特子空间定理的主要结果的无条件形式.
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On non-Zariski density of (D,S)-integral points in forward orbits and the Subspace Theorem

Working over a base number field K, we study the attractive question of Zariski non-density for (D,S)-integral points in Of(x) the forward f-orbit of a rational point xX(K). Here, f:XX is a regular surjective self-map for X a geometrically irreducible projective variety over K. Given a non-zero and effective f-quasi-polarizable Cartier divisor D on X and defined over K, our main result gives a sufficient condition, that is formulated in terms of the f-dynamics of D, for non-Zariski density of certain dynamically defined subsets of Of(x). For the case of (D,S)-integral points, this result gives a sufficient condition for non-Zariski density of integral points in Of(x). Our approach expands on that of Yasufuku, [13], building on earlier work of Silverman [11]. Our main result gives an unconditional form of the main results of [13]; the key arithmetic input to our main theorem is the Subspace Theorem of Schmidt in the generalized form that has been given by Ru and Vojta in [10] and expanded upon in [3] and [6].

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