Locally nilpotent derivations with a PID ring of constants

Abstract:

Let $\mathcal {K}$ be a commutative field of characteristic zero, $\mathcal {A}$ be a domain containing $\mathcal {K}$ and $\partial$ be a locally nilpotent $\mathcal {K}$-derivation of $\mathcal {A}$. We give in this paper a description of the differential $\mathcal {K}$-algebra $(\mathcal {A},\partial )$ under the assumptions that the ring of constants $\mathcal {A}^{\partial }$ of $\partial$ is a PID, $\partial$ is fixed point free and its special fibers are reduced.