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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2412.04750 |
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| _version_ | 1866909417898770432 |
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| author | Huang, Wenyong Zhang, Xiang |
| author_facet | Huang, Wenyong Zhang, Xiang |
| contents | Prelle and Singer showed in 1983 that if a system of ordinary differential equations defined on a differential field $K$ has a first integral in an elementrary field extension $L$ of $K$, then it must have a first integral consisting of algebraic elements over $K$ via their constant powers and logarithms. Based on this result they further proved that an elementary integrable planar polynomial differential system has an integrating factor which is a fractional power of a rational function. Here we extend their results and prove that any $n$ dimensional elementary integrable polynomial vector field has $n-1$ functionally independent first integrals being composed of algebraic elements over $K$. Furthermore, using the Galois theory we prove that the vector field has a rational Jacobian multiplier. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_04750 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Reduction of Elementary Integrability of Polynomial Vector Fields Huang, Wenyong Zhang, Xiang Dynamical Systems Classical Analysis and ODEs Group Theory 34A34, 37C10, 34C14, 37G05 Prelle and Singer showed in 1983 that if a system of ordinary differential equations defined on a differential field $K$ has a first integral in an elementrary field extension $L$ of $K$, then it must have a first integral consisting of algebraic elements over $K$ via their constant powers and logarithms. Based on this result they further proved that an elementary integrable planar polynomial differential system has an integrating factor which is a fractional power of a rational function. Here we extend their results and prove that any $n$ dimensional elementary integrable polynomial vector field has $n-1$ functionally independent first integrals being composed of algebraic elements over $K$. Furthermore, using the Galois theory we prove that the vector field has a rational Jacobian multiplier. |
| title | Reduction of Elementary Integrability of Polynomial Vector Fields |
| topic | Dynamical Systems Classical Analysis and ODEs Group Theory 34A34, 37C10, 34C14, 37G05 |
| url | https://arxiv.org/abs/2412.04750 |