Darboux theory of integrability for real polynomial vector fields on the $n-$dimensional ellipsoid
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929563995471872 |
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| author | Llibre, J. Murza, Adrian C. |
| author_facet | Llibre, J. Murza, Adrian C. |
| contents | We extend to the $n$-dimensional ellipsoid contained in $\R^{n+1},$ the Darboux theory of integrability for polynomial vector fields in the $n$-dimensional sphere (Llibre et al., 2018). New results on the maximum number of invariant parallels and meridians of polynomial vector fields $\X$ on the invariant $n-$dimensional ellipsoid, as a function of its degree, are provided. Our results extend the known result on the upper bound for the number of invariant hyperplanes that a polynomial vector field $\Y$ in $\R^n$ can have in function of the degree of $\Y$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_21336 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Darboux theory of integrability for real polynomial vector fields on the $n-$dimensional ellipsoid Llibre, J. Murza, Adrian C. Dynamical Systems 34C07, 34C05, 34C40 We extend to the $n$-dimensional ellipsoid contained in $\R^{n+1},$ the Darboux theory of integrability for polynomial vector fields in the $n$-dimensional sphere (Llibre et al., 2018). New results on the maximum number of invariant parallels and meridians of polynomial vector fields $\X$ on the invariant $n-$dimensional ellipsoid, as a function of its degree, are provided. Our results extend the known result on the upper bound for the number of invariant hyperplanes that a polynomial vector field $\Y$ in $\R^n$ can have in function of the degree of $\Y$. |
| title | Darboux theory of integrability for real polynomial vector fields on the $n-$dimensional ellipsoid |
| topic | Dynamical Systems 34C07, 34C05, 34C40 |
| url | https://arxiv.org/abs/2410.21336 |