A new measure of a partial differential field extension

Israel Zuckerman

Vol. 15 (1965), No. 1, 357–371

DOI: 10.2140/pjm.1965.15.357

Abstract

Let $G$ be a differential field of characteristic zero with the commuting derivations $d_1,\cdots ,d_m$. If $F$ is a differential subfield of $G$, the algebraic and differential degrees of transcendence of $G$ over $F$, denoted respectively by $d\left(G∕F\right)$ and $d.d\left(G∕F\right)$ are numerical invariants of the extension. Unlike the ordinary differential case $\left(m=1\right)$ $d.d.\left(G∕F\right)=0$ does not imply that $d\left(G∕F\right)$ is finite. In this paper an intermediate measure of the extension is constructed, called the limit vector. The first and last components of this vector correspond to $d.d\left(G∕F\right)$ and $d\left(G∕F\right)$ respectively, and the limit vector is additive.

Mathematical Subject Classification

Primary: 12.80

Milestones

Received: 23 January 1964

Published: 1 March 1965