On Certain Towers of Extensions by Antiderivatives

Abstract

Let F be a characteristic zerodifferential field with an algebraically closed field of constantsC . Let E and K be no new constants extensions of F, E contains K, Kis an extension by antiderivatives of Fand Econtain antiderivatives y1,&hellip,yn of K. Theantiderivatives y1,&hellip,ynof K are called J-I-Eantiderivatives if the derivative of each yisatisfies certain conditions. We will provide a new proof for the Kolchin-Ostrowski theorem andgeneralize this theorem for a tower of extensions by J-I-Eantiderivatives and use this generalized version of the theorem toclassify the finitely differentially generated subfields of thistower. In the process, we will show that the J-I-E antiderivativesare algebraically independent over the ground differential field.An example of a J-I-E tower is the iterated antiderivative extensionsof the field of rational functions C(x) generated by iteratedlogarithms, closed at each stage by all (translation)automorphisms. We analyze the algebraic and differential structureof these extensions. In particular, we show that the nth iteratedlogarithms and their translates are algebraically independent overthe field generated by all lower level iterated logarithms. Ouranalysis provides an algorithm for determining the differentialfield generated by any rational expression in iterated logarithms.