Liouvillian integrability of vector fields in higher dimensions
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909967653535744 |
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| author | Aziz, Waleed Christopher, Colin Pantazi, Chara Walcher, Sebastian |
| author_facet | Aziz, Waleed Christopher, Colin Pantazi, Chara Walcher, Sebastian |
| contents | We consider complex rational vector fields in dimension $n>2$ (equivalently, differential forms of degree $n-1$ in $n$ variables) which admit a Liouvillian first integral. Extending a classical result by Singer for $n=2$, our main result states that there exists a first integral which is obtained by two successive integrations from one-forms with coefficients in a finite algebraic extension of the rational function field. The proof uses Puiseux series in a novel way to simplify computations. We also apply this method to give elementary proofs of Singer's theorem for rational one-forms, and of the Prelle-Singer theorem on elementary integrability of rational vector fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_15522 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Liouvillian integrability of vector fields in higher dimensions Aziz, Waleed Christopher, Colin Pantazi, Chara Walcher, Sebastian Exactly Solvable and Integrable Systems Classical Analysis and ODEs 34A99, 12H05, 34M15 We consider complex rational vector fields in dimension $n>2$ (equivalently, differential forms of degree $n-1$ in $n$ variables) which admit a Liouvillian first integral. Extending a classical result by Singer for $n=2$, our main result states that there exists a first integral which is obtained by two successive integrations from one-forms with coefficients in a finite algebraic extension of the rational function field. The proof uses Puiseux series in a novel way to simplify computations. We also apply this method to give elementary proofs of Singer's theorem for rational one-forms, and of the Prelle-Singer theorem on elementary integrability of rational vector fields. |
| title | Liouvillian integrability of vector fields in higher dimensions |
| topic | Exactly Solvable and Integrable Systems Classical Analysis and ODEs 34A99, 12H05, 34M15 |
| url | https://arxiv.org/abs/2512.15522 |