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摘要翻译：
对于Z_p^n的一个可定义子集（或Z_p上的有限型格式），人们可以以自然的方式关联一棵树。已知对应的Poincare级数P(X)=\sum_in_ix^i是有理的，其中N_i是树在深度i处的节点数。这表明树木本身远不是任意的。我们对这类可能树提出了一个猜想的、纯组合的描述，并为它提供了一些证据。我们证明了我们类中的任何树确实来自可定义集，并证明了在弱光滑性假设下，对于Z_p^2的可定义子集，以及对于一维集，可定义集（或方案）的树存在于我们类中。
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英文标题：
《Trees of definable sets over the p-adics》
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作者：
Immanuel Halupczok
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最新提交年份：
2008
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分类信息：
一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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一级分类：Mathematics 数学
二级分类：Logic 逻辑
分类描述：Logic, set theory, point-set topology, formal mathematics
逻辑，集合论，点集拓扑，形式数学
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英文摘要：
To a definable subset of Z_p^n (or to a scheme of finite type over Z_p) one can associate a tree in a natural way. It is known that the corresponding Poincare series P(X) = \sum_i N_i X^i is rational, where N_i is the number of nodes of the tree at depth i. This suggests that the trees themselves are far from arbitrary. We state a conjectural, purely combinatorial description of the class of possible trees and provide some evidence for it. We verify that any tree in our class indeed arises from a definable set, and we prove that the tree of a definable set (or of a scheme) lies in our class in three special cases: under weak smoothness assumptions, for definable subsets of Z_p^2, and for one-dimensional sets.
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PDF链接：
https://arxiv.org/pdf/0806.4469
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