Difference "abc" theorem for entire functions and Difference analogue of truncated version of Nevanlinna second main theorem
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866915079510818816 |
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| author | Chen, Rui-Chun Wen, Zhi-Tao |
| author_facet | Chen, Rui-Chun Wen, Zhi-Tao |
| contents | In this paper, we focus on the difference analogue of the Stothers-Mason theorem for entire functions of order less than 1, which can be seen as difference $abc$ theorem for entire functions. We also obtain the difference analogue of truncated version of Nevanlinna second main theorem which reveals that a subnormal meromorphic function $f(z)$ such that $Δf(z)\not\equiv 0$ cannot have too many points with long length in the complex plane. Both theorems depend on new definitions of the length of poles and zeros of a given meromorphic function in a domain. As for the application, we consider entire solutions of difference Fermat functional equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_06317 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Difference "abc" theorem for entire functions and Difference analogue of truncated version of Nevanlinna second main theorem Chen, Rui-Chun Wen, Zhi-Tao Complex Variables Number Theory 30D35, 30D20 In this paper, we focus on the difference analogue of the Stothers-Mason theorem for entire functions of order less than 1, which can be seen as difference $abc$ theorem for entire functions. We also obtain the difference analogue of truncated version of Nevanlinna second main theorem which reveals that a subnormal meromorphic function $f(z)$ such that $Δf(z)\not\equiv 0$ cannot have too many points with long length in the complex plane. Both theorems depend on new definitions of the length of poles and zeros of a given meromorphic function in a domain. As for the application, we consider entire solutions of difference Fermat functional equations. |
| title | Difference "abc" theorem for entire functions and Difference analogue of truncated version of Nevanlinna second main theorem |
| topic | Complex Variables Number Theory 30D35, 30D20 |
| url | https://arxiv.org/abs/2405.06317 |