Riso-分层与一个树不变量
Riso-stratifications and a tree invariant
AI总结：
本文提出典范的 riso-分层及 riso-tree 奇点不变量，证明其分层在相应一阶语言中可定义，并应用于局部动机 Poincaré 级数。
摘要：
我们引入了一种新的分层概念（“riso-stratification”），它是典范的，并且存在于多种环境中，包括不同的拓扑域，如 $\mathbb{C}$、$\mathbb{R}$ 和 $\mathbb{Q}_p$，也包括 $\mathbb{R}$ 上不同的 o-minimal 结构。Riso-分层直接依据沿分层的适当平凡性概念来定义；关键困难和主要结果在于，以这种方式定义的分层“本质上是代数的”，即在相应的一阶语言中可定义。作为示例应用，我们证明局部动机 Poincaré 级数在某种意义上沿 riso-分层的分层是平凡的。在 riso-分层概念背后是一个新的奇点不变量，我们称之为“riso-tree”，它以典范方式捕捉了原先包含在 Lipschitz 分层的非典范分层中的信息。在推进 Poincaré 级数应用的过程中，我们除其他结果外还证明，我们的概念与动机积分能够良好地相互作用。
英文摘要：
We introduce a new notion of stratification (``riso-stratification''), which is canonical and which exists in a variety of settings, including different topological fields like $\mathbb{C}$, $\mathbb{R}$ and $\mathbb{Q}_p$, and also including different o-minimal structures on $\mathbb{R}$. Riso-stratifications are defined directly in terms of a suitable notion of triviality along strata; the key difficulty and main result is that the strata defined in this way are ``algebraic in nature'', i.e., definable in the corresponding first-order language. As an example application, we show that local motivic Poincaré series are, in some sense, trivial along the strata of the riso-stratification. Behind the notion of riso-stratification lies a new invariant of singularities, which we call the ``riso-tree'', and which captures, in a canonical way, information that was contained in the non-canonical strata of a Lipschitz stratification. On our way to the Poincaré series application, we show, among others, that our notions interact well with motivic integration.
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