On the holomorphic foliations admitting a common invariant algebraic set

Abstract: 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.