A few remarks on the Generalized Vanishing Conjecture

We show that the Generalized Vanishing Conjecture $${\mathrm{\forall }}_{m\ge 1}[{\text{Lam}}^m{f}^m=0]⟹{\mathrm{\forall }}_{m\gg 0}[{\text{Lam}}^m(g{f}^m)=0]$$ for a fixed differential operator $\text{Lam}\in k[\mathrm{\partial }]$ follows from a special case of it, namely that the additional factor $g$ is a power of the radical polynomial $f$. Next we show that in order to prove the Generalized Vanishing Conjecture (up to some bound on the degree of $\text{Lam}$), we may assume that $\text{Lam}$ is a linear combination of powers of distinct partial derivatives. At last, we show that the Generalized Vanishing Conjecture holds for products of linear forms in $\mathrm{\partial }$, in particular homogeneous differential operators $\Lambda \in k[\partial_1,\partial_2]$.