Riso-stratifications and a tree invariant

TL;DR

This paper introduces riso-stratifications, a canonical stratification concept applicable across various fields and structures, defining an algebraic invariant called the riso-tree that captures singularity information and interacts with motivic integration.

Contribution

It presents the novel notion of riso-stratification, proves its algebraic definability, and introduces the riso-tree invariant, extending stratification theory to new settings.

Findings

Riso-stratifications are definable in first-order languages.

Local motivic Poincaré series are trivial along riso-stratification strata.

The riso-tree captures singularity information in a canonical way.

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 Poincar\'e 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 Poincar\'e series application, we show, among others, that our notions interact well with motivic integration.