On the holomorphic foliations admitting a common invariant algebraic set
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913919755354112 |
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| author | Dong, Guangfeng Lu, Chujun |
| author_facet | Dong, Guangfeng Lu, Chujun |
| contents | In this paper, we study the holomorphic foliations admitting a common invariant algebraic set $C$ defined by a polynomial $f$ in $ \mathbb{K}[x_1,x_2,...,x_n]$ over any characteristic $0$ subfield $\mathbb{K}\subseteq\mathbb{C}$. For the $\mathbb{K}[x_1,x_2,...,x_n]$-module $V_f$ of vector fields generating foliations that admit $C$ as an invariant set, we provide several conditions under which the module $V_f$ can be freely generated by a minimal generating set. In particular, when $n=2$ and $f$ is a weakly tame polynomial, we show that the $\mathbb{K}[x,y]$-module $V_f$ is freely generated by two polynomial vector fields, one of which is the Hamiltonian vector field induced by $f$, if and only if, $f$ belongs to the Jacobian ideal $\langle f_x, f_y\rangle$ in $\mathbb{K}[x,y]$. Our proof employs a purely elementary method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_11172 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the holomorphic foliations admitting a common invariant algebraic set Dong, Guangfeng Lu, Chujun Dynamical Systems Primary: 37F75, Secondary: 32M25, In this paper, we study the holomorphic foliations admitting a common invariant algebraic set $C$ defined by a polynomial $f$ in $ \mathbb{K}[x_1,x_2,...,x_n]$ over any characteristic $0$ subfield $\mathbb{K}\subseteq\mathbb{C}$. For the $\mathbb{K}[x_1,x_2,...,x_n]$-module $V_f$ of vector fields generating foliations that admit $C$ as an invariant set, we provide several conditions under which the module $V_f$ can be freely generated by a minimal generating set. In particular, when $n=2$ and $f$ is a weakly tame polynomial, we show that the $\mathbb{K}[x,y]$-module $V_f$ is freely generated by two polynomial vector fields, one of which is the Hamiltonian vector field induced by $f$, if and only if, $f$ belongs to the Jacobian ideal $\langle f_x, f_y\rangle$ in $\mathbb{K}[x,y]$. Our proof employs a purely elementary method. |
| title | On the holomorphic foliations admitting a common invariant algebraic set |
| topic | Dynamical Systems Primary: 37F75, Secondary: 32M25, |
| url | https://arxiv.org/abs/2505.11172 |