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

We study the LND conjecture concerning the images of locally nilpotent derivations, which arose from the Jacobian conjecture. Let $R$ be a domain containing a field of characteristic zero. We prove that, when $R$ is a one-dimensional unique factorization domain, the image of any locally nilpotent $R$-derivation of the bivariate polynomial algebra $R[x,y]$ is a Mathieu-Zhao subspace. Moreover, we prove that, when $R$ is a Dedekind domain, the image of a locally nilpotent $R$-derivation of $R[x,y]$ with some additional conditions...

Let $k$ be a field of characteristic zero and $B$ a $k$-domain. Let $R$ be a retract of $B$ being the kernel of a locally nilpotent derivation of $B$. We show that if $B=R\oplus I$ for some principal ideal $I$ (in particular, if $B$ is a UFD), then $B=R^{\left[1\right]}$, i.e., $B$ is a polynomial algebra over $R$ in one variable. It is natural to ask that, if a retract $R$ of a $k$-UFD $B$ is the kernel of two commuting locally nilpotent derivations of $B$, then does it follow that $B\cong R^{\left[2\right]}$? We give a negative answer to this question. The interest in retracts comes...