A case of the generalized vanishing conjecture

TL;DR

This paper proves the generalized vanishing conjecture for a specific class of differential operators and all polynomials, contributing to the broader understanding related to the Jacobian conjecture.

Contribution

It establishes the validity of the GVC for the operator b3=(b4_x-a(b4_y))b4_y and all polynomials, advancing the theoretical framework of the conjecture.

Findings

GVC holds for the operator b3 and all polynomials.

Supports the study of the Jacobian conjecture.

Provides a new case where GVC is verified.

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.