Riso-stratifications and a tree invariant

TL;DR

A new notion of stratification (“riso-stratifications”) is introduced, which is entirely canonical and which exists in a variety of settings, including different topological fields like C, R and Qp, and also including different o-minimal structures on R.

Abstract

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.

Figures (6)

• Figure 1: On the left hand side, the intersection of some set $\mathbf{X}(K)$ with some infinitesimal neighbourhood $U_x$, and on the right hand side, its preimage under a risometry $\varphi$ making it horizontally translation invariant. That $\varphi$ is a risometry means that the differences $y_1 - y_2$ and $\varphi(y_1) - \varphi(y_2)$ are almost equal.
• Figure 2: The set $\mathbf{X}(K) \subseteq K^3$ given by $z^2 + y^2 = x^3$ shows that the riso-tree captures more information than the riso-stratification.
• Figure 3: The risometry in Example \ref{['exa:vert:transl']} turns a straight line segment into the graph of a function $f$, by translating things vertically.
• Figure 4: The ingredients of Claim \ref{['claim.ind.UVW.new']} (2): $(\mathbf{S}_{i}(K))_i$ is only $\bar{W}$-riso-trivial, due do $\mathbf{S}_{\dim \bar{W}}(K)$ intersecting $\mathbf{B}(K)$, but $\chi_K$ might nevertheless also be $\bar{U}$-riso-trivial. To check this, one needs to verify whether ${\boldsymbol{\chi}}_K |_{B_1}$ and ${\boldsymbol{\chi}}_K|_{B_2}$ are in risometry for every pair of balls $B_1$ and $B_2$ as in the claim.
• Figure 5: Proving (**): Using induction over $\kappa$, we obtain $\bar{U}$-riso-triviality on stripes $D_0$ containing $\kappa$ elements of $\mathbf{S}_\ell(K) \cap F$. The case $\kappa = 0$ is obtained from an outer induction over $\ell$.

Theorems & Definitions (173)

• Definition 2.1.10: $\mathbf{RV}^{(n)}$
• Definition 2.1.12: Balls
• Definition 2.1.14: Dimension
• Example 2.2.2
• Remark 2.2.3
• Proposition 2.2.4: iCR.hmin
• Lemma 2.2.5
• Theorem 2.2.6: iBW.sph
• Remark 2.2.7
• Definition 2.3.1