2022/06/07 by David Bradley-Williams, Bradley-Williams, David, Immanuel Halupczok +1 · 1 citation
Computer Science · #Advanced Algebra and Logic
paper · pdf · doi:10.48550/arxiv.2206.03438
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 ℂ, ℝ and ℚp, and also including different o-minimal structures on 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.