A case of the generalized vanishing conjecture

Abstract

In this paper, we show that the GVC (generalized vanishing conjecture) holds for the differential operator $\Lambda =({\partial }_x-\Phi ({\partial }_y)){\partial }_y$ and all polynomials $P(x,y)$, where $\Phi (t)$ is any polynomial over the base field. The GVC arose from the study of the Jacobian conjecture.