Riso-stratifications and a tree invariant
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2022
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| _version_ | 1866916098662727680 |
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| author | Bradley-Williams, David Halupczok, Immanuel |
| author_facet | Bradley-Williams, David Halupczok, Immanuel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_03438 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Riso-stratifications and a tree invariant Bradley-Williams, David Halupczok, Immanuel Algebraic Geometry Logic 03C60, 03C65, 03C98, 12J25, 32S60, 03H05, 14B05, 14B20, 14G20 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. |
| title | Riso-stratifications and a tree invariant |
| topic | Algebraic Geometry Logic 03C60, 03C65, 03C98, 12J25, 32S60, 03H05, 14B05, 14B20, 14G20 |
| url | https://arxiv.org/abs/2206.03438 |