Locally Nilpotent Derivations over a UFD and an Application to Rank Two Locally Nilpotent Derivations ofk[X1,…,Xn]

Given a UFDRcontaining the rational numbers, we study locally nilpotentR-derivations of the polynomial ringR[X, Y]; in particular, we give a generalization of Rentschlerʹs Theorem and a criterion for the existence of a slice. These results are then applied to describe rank two locally nilpotent derivations ofk[X1,…,Xn], wherekis a field of characteristic zero. We also give an example of a nontriangulable locally nilpotent derivation ofk[X, Y, Z] whose set of fixed points is a line.