Conservative and divergence free algebraic vector fields

Abstract:

Suppose $k$ is a field of characteristic 0 and ${k^{[n]}} = k[{x_1}, \ldots ,{x_n}]$. If ${u_i}$, ${f_j} \in {k^{[n]}}$ for $1 \leqslant i \leqslant n$, $1 \leqslant j \leqslant m$, $u = ({u_1}, \ldots ,{u_n})$, the ${f_j}$ are relatively prime, and each ${f_j}u$ is conservative, then $u$ is conservative and $({f_1}, \ldots ,{f_m})$ is unimodular. Given any $u$ with $\left | {J(u)} \right | = 1$, then each derivation $\partial /\partial {u_i}$, has divergence 0. If $D:{k^{[n]}} \to {k^{[n]}}$ is a $k$-derivation with kernel of dimension $n$ - $- 1$, then there exists a $g$ so that $gD$ has divergence 0.