On divergence and sums of derivations

Abstract

Let \(K\) be an algebraically closed   field of characteristic zero and \(A\)  a field of algebraic functions in \(n\) variables over \(\mathbb K\). (i.e. \(A\) is a finite dimensional algebraic extension of the field \(\mathbb K(x_1, \ldots, x_n)\) ). If \(\textit{D}\) is a \(\mathbb K\)-derivation of \(A\), then its divergence \(div \textit{D}\) is an important geometric  characteristic of  \(\textit{D}\) (\(\textit{D}\) can be considered as a vector field with coefficients in \(A\)). A relation between expressions of \(div \textit{D}\) in different transcendence bases of   \(A\) is pointed out. It is also proved that every divergence-free derivation \(\textit{D}\) on the polynomial ring \(\mathbb K[x, y, z]\) is a sum of at most two jacobian derivation.