We introduce a new notion of stratification (“riso-stratifications”), 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.
