On definable groups in dp-minimal topological fields equipped with a generic derivation
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915283007963136 |
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| author | Point, Françoise |
| author_facet | Point, Françoise |
| contents | Let $T$ be a complete, model-complete, geometric dp-minimal $\mathcal{L}$-theory of topological fields of characteristic $0$ and let $T(\partial)$ be the theory of expansions of models of $T$ by a derivation $\partial$. We assume that $T(\partial)$ has a model-companion $T_{\partial}$. Let $Γ$ be a finite-dimensional $\mathcal{L}_\partial$-definable group in a model of $T_\partial$. Then we show that $Γ$ densely and definably embeds in an $\mathcal{L}$-definable group $G$.
Further, using a $C^1$-cell decomposition result, we show that $Γ$ densely and definably embeds in a definable $D$-group, generalizing the classical construction of Buium of algebraic $D$-groups and extending for that class of fields, results obtained in arXiv:2208.08293, arXiv:2305.16747. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07044 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On definable groups in dp-minimal topological fields equipped with a generic derivation Point, Françoise Logic 03C60, 12H05, 12L12 Let $T$ be a complete, model-complete, geometric dp-minimal $\mathcal{L}$-theory of topological fields of characteristic $0$ and let $T(\partial)$ be the theory of expansions of models of $T$ by a derivation $\partial$. We assume that $T(\partial)$ has a model-companion $T_{\partial}$. Let $Γ$ be a finite-dimensional $\mathcal{L}_\partial$-definable group in a model of $T_\partial$. Then we show that $Γ$ densely and definably embeds in an $\mathcal{L}$-definable group $G$. Further, using a $C^1$-cell decomposition result, we show that $Γ$ densely and definably embeds in a definable $D$-group, generalizing the classical construction of Buium of algebraic $D$-groups and extending for that class of fields, results obtained in arXiv:2208.08293, arXiv:2305.16747. |
| title | On definable groups in dp-minimal topological fields equipped with a generic derivation |
| topic | Logic 03C60, 12H05, 12L12 |
| url | https://arxiv.org/abs/2505.07044 |