On image ideals of nice and quasi-nice derivations over a UFD
Nikhilesh Dasgupta, Animesh Lahiri
TL;DR
This work determines explicit generators for the image ideals of irreducible nice and quasi-nice locally nilpotent derivations on two-variable polynomial rings over a UFD. By developing LND filtrations and top-degree (tilde) ideals, the authors reduce the problem to algebraic data in the kernel $A={ m Ker}(D)$ and derive concrete formulas: for nice $D$ with $D^2X_1=D^2X_2=0$, $I_j={(DX_1,DX_2)}^jA$, while for non-fixed-point-free strictly $1$-quasi-nice $D$, $I_j$ factors as $b^{m_j}ar{I_j}$ with explicitly described exponents and contained ideals. The results extend to PID/DVR settings and include a detailed quasi-nice case with a refined stratification of higher image ideals, accompanied by examples and conjectures guiding future work. The methods provide concrete, computable generators for image ideals, contributing to understanding the structure of LNDs and supporting related conjectures in the area. Overall, the paper offers a practical framework to compute image ideals in $R^{[2]}$ for a broad class of derivations, with implications for the Freeness conjecture landscape and related kernel-structure questions.
Abstract
In this paper, for a field $k$ of characteristic zero and a finitely generated $k$-algebra $R$, we give a set of generators for the image ideals of irreducible nice and quasi-nice $R$-derivations on the polynomial ring $R[X,Y]$, where $R$ is a UFD.