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 K. (i.e. A is a finite dimensional algebraic extension of the field K(x1, . . . , xn) ). If D is a K-derivation of A, then its divergence divD is an important geometric characteristic of D (D canbe considered as a vector field with coefficients in A). A relation between expressions of divD in different transcendence bases of A is pointed out. It is also proved that every divergence-free derivation D on the polynomial ring K[x, y, z] is a sum of at most two jacobian derivation.