On holomorphicity of Hartogs series satisfying algebraic relations
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916938417963008 |
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| author | Aoki, Hiroki Saito, Kyoji |
| author_facet | Aoki, Hiroki Saito, Kyoji |
| contents | We consider a formal power series in one variable whose coefficients are holomorphic functions in a given multidimensional complex domain. Assume the following two conditions on the series. (C1) The restriction of the series at each point of a dense subset of the domain converges in an open disk of a fixed radius. (C2) The series is algebraic over the ring of holomophic functions on the direct product space of the domain and the disk. The main theorem of the present note is that the series defines a holomorphic function on the direct product space. We also give an example where the condition (C2) is essentially necessary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_10641 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On holomorphicity of Hartogs series satisfying algebraic relations Aoki, Hiroki Saito, Kyoji Complex Variables primary: 32A05, secondary: 32A10, 32D15 We consider a formal power series in one variable whose coefficients are holomorphic functions in a given multidimensional complex domain. Assume the following two conditions on the series. (C1) The restriction of the series at each point of a dense subset of the domain converges in an open disk of a fixed radius. (C2) The series is algebraic over the ring of holomophic functions on the direct product space of the domain and the disk. The main theorem of the present note is that the series defines a holomorphic function on the direct product space. We also give an example where the condition (C2) is essentially necessary. |
| title | On holomorphicity of Hartogs series satisfying algebraic relations |
| topic | Complex Variables primary: 32A05, secondary: 32A10, 32D15 |
| url | https://arxiv.org/abs/2411.10641 |