A few remarks on the Generalized Vanishing Conjecture

TL;DR

This paper investigates the Generalized Vanishing Conjecture, demonstrating its reduction to special cases, simplifying assumptions on the differential operator, and proving it for products of linear forms in partial derivatives.

Contribution

It shows the conjecture follows from a special case, reduces the operator to linear combinations of powers, and proves it for products of linear forms.

Findings

Conjecture follows from a case where g is a power of f

Reduction to operators as linear combinations of powers of derivatives

Proven for products of linear forms in partial derivatives

Abstract

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