Exponential periods and o-minimality
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2020
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| _version_ | 1866908287454150656 |
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| author | Commelin, Johan Habegger, Philipp Huber, Annette |
| author_facet | Commelin, Johan Habegger, Philipp Huber, Annette |
| contents | Let $α\in \mathbb{C}$ be an exponential period. We show that the real and imaginary part of $α$ are up to signs volumes of sets definable in the o-minimal structure generated by $\mathbb{Q}$, the real exponential function and ${\sin}|_{[0,1]}$. This is a weaker analogue of the precise characterisation of ordinary periods as numbers whose real and imaginary part are up to signs volumes of $\mathbb{Q}$-semi-algebraic sets. Furthermore, we define a notion of naive exponential periods and compare it to the existing notions using cohomological methods. This points to a relation between the theory of periods and o-minimal structures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2007_08280 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Exponential periods and o-minimality Commelin, Johan Habegger, Philipp Huber, Annette Number Theory 11G35, 14F25, 14F40, 14P10, 03C64 Let $α\in \mathbb{C}$ be an exponential period. We show that the real and imaginary part of $α$ are up to signs volumes of sets definable in the o-minimal structure generated by $\mathbb{Q}$, the real exponential function and ${\sin}|_{[0,1]}$. This is a weaker analogue of the precise characterisation of ordinary periods as numbers whose real and imaginary part are up to signs volumes of $\mathbb{Q}$-semi-algebraic sets. Furthermore, we define a notion of naive exponential periods and compare it to the existing notions using cohomological methods. This points to a relation between the theory of periods and o-minimal structures. |
| title | Exponential periods and o-minimality |
| topic | Number Theory 11G35, 14F25, 14F40, 14P10, 03C64 |
| url | https://arxiv.org/abs/2007.08280 |