Algebraic Independence of generic Painlevé Transcendents: P_III and P_VI
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arXiv
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| Format: | Preprint |
| Publié: |
2017
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| _version_ | 1866917270454796288 |
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| author | Nagloo, Joel |
| author_facet | Nagloo, Joel |
| contents | We prove that if y"=f(y,y',t) is a generic Painlevé equation from the class III and VI, and if y_1,...,y_n are distinct solutions, then y_1,y_1',...,y_n,y_n' are algebraically independent over C(t). This improves the weaker results obtained by the author and Pillay and completely prove the algebraic independence conjecture for the generic Painlevé transcendents. In the process, we also prove that any three distinct solutions of a Riccati equation are algebraic independent over C(t), provided that there are no solutions in the algebraic closure of C(t). This answers a very natural question in the theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1708_02962 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Algebraic Independence of generic Painlevé Transcendents: P_III and P_VI Nagloo, Joel Algebraic Geometry Logic 14H05, 14H70, 34M55, 03C60 We prove that if y"=f(y,y',t) is a generic Painlevé equation from the class III and VI, and if y_1,...,y_n are distinct solutions, then y_1,y_1',...,y_n,y_n' are algebraically independent over C(t). This improves the weaker results obtained by the author and Pillay and completely prove the algebraic independence conjecture for the generic Painlevé transcendents. In the process, we also prove that any three distinct solutions of a Riccati equation are algebraic independent over C(t), provided that there are no solutions in the algebraic closure of C(t). This answers a very natural question in the theory. |
| title | Algebraic Independence of generic Painlevé Transcendents: P_III and P_VI |
| topic | Algebraic Geometry Logic 14H05, 14H70, 34M55, 03C60 |
| url | https://arxiv.org/abs/1708.02962 |