Submission date: 7 June 2022

Abstract:

We introduce a new notion of stratification (“riso-stratifications”), which is entirely canonical and which exists in a variety of settings, including different topological fields like ℂ, ℝ and ℚ_p, 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 additional information not visible in the riso-stratification. As for the riso-stratifications, our main result about riso-trees is that they are definable. On our way to the Poincaré series application, we also show that our notions interact well with motivic integration.

Mathematics Subject Classification: 03C60, 03C65, 03C98, 12J25, 32S60, 03H05, 14B05, 14B20, 14G20

Keywords and phrases:

Full text arXiv 2206.03438: pdf, ps.