RESEARCH ARTICLE

On divergence and sums of derivations

E. Chapovsky, O. Shevchyk

Department of Algebra and Mathematical Logic, Faculty of Mechanics and Mathematics, Kyiv Taras Shevchenko University, 64, Volodymyrska street, 01033 Kyiv, Ukraine

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 $D$ 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.

Keywords: polynomial ring, derivation, divergence, jacobian derivation, transcendence basis.

MSC: Primary 13N15; Secondary 13A99, 17B66

Received: 05.12.2016
Revised: 07.12.2016

Language: English

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