Differential field extensions with no movable algebraic branches (Q1355156)

The singular points play an important role in the classic theory of algebraic differential equations [see \textit{A. R. Forsyth}, Theory of differential equations, Cambridge University Press (1890; JFM 22.0347.02)]. The differential algebraic analogies of singular points are discussed in the paper. Differential field extensions with no movable algebraic branches are defined, and such a differential field extension is proved to be a Painleve-Umemura extension provided that it is included in a decomposable extension, which was defined previously by the author.