Locally nilpotent derivations of polynomial rings (Q1916626)

This note is concerned with locally nilpotent derivations, that are derivations \(D\) on a ring \(R\) fulfilling \(\text{Nil} (D)=R\), \(\text{Nil} (D)\) being the set \(\{x\in R; \exists n\in \mathbb{N}\) with \(D^n(x)=0\}\). In particular the following theorem is shown: Let \(F\) be an irreducible algebraic curve in \(\mathbb{A}^2\) given by the equation \(f(x,y)=0\). Then \(F\cong \mathbb{A}^1\) if and only if \(D_f=f_y {\partial \over \partial x} - f_x{\partial \over \partial y}\) is a locally nilpotent derivation of the ring \(k[x,y]\); here \(\mathbb{A}^n\) denotes the \(n\)-dimensional affine space over an algebraically closed field \(k\) with \(\text{char} k=0\). Also the rectifiability of a surface \(G\subset \mathbb{A}^3\) is expressed in terms of locally nilpotent derivations.