On divergence and sums of derivations

E. Chapovsky ; O. Shevchyk

Algebra and discrete mathematics, Tome 24 (2017) no. 1, pp. 99-105

Résumé

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 $D$ is a $\mathbb K$-derivation of $A$, then its divergence $\operatorname{div} D$ is an important geometric characteristic of $D$ ($D$ can be considered as a vector field with coefficients in $A$). A relation between expressions of $\operatorname{div} D$ in different transcendence bases of $A$ is pointed out. It is also proved that every divergence-free derivation $D$ on the polynomial ring $\mathbb K[x, y, z]$ is a sum of at most two jacobian derivation.

Détail

Keywords: polynomial ring, derivation, jacobian derivation, transcendence basis.
Mots-clés : divergence