A few remarks on the Generalized Vanishing Conjecture

We show that the Generalized Vanishing Conjecture $$forall_{m ge 1} [Lambda^m f^m = 0] Longrightarrow forall_{m gg 0}[Lambda^m (g f^m) = 0]$$ for a fixed differential operator ${Lambda in k[partial]}$ 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 Λ), we may assume that Λ 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 ?, in particular homogeneous differential operators ${Lambda in k[partial_1,partial_2]}$ .