On Certain Towers of Extensions by Antiderivatives

Abstract

Let F be a characteristic zero

differential field with an algebraically closed field of constants

C . Let E and K be no new constants extensions of F, E contains K, Kis an extension by antiderivatives of F

and Econtain antiderivatives y1,&hellip,yn of K. The

antiderivatives y1,&hellip,ynof K are called J-I-E

antiderivatives if the derivative of each yisatisfies certain conditions. We will provide a new proof for the Kolchin-Ostrowski theorem and

generalize this theorem for a tower of extensions by J-I-E

antiderivatives and use this generalized version of the theorem to

classify the finitely differentially generated subfields of this

tower. In the process, we will show that the J-I-E antiderivatives

are algebraically independent over the ground differential field.

An example of a J-I-E tower is the iterated antiderivative extensions

of the field of rational functions C(x) generated by iterated

logarithms, closed at each stage by all (translation)

automorphisms. We analyze the algebraic and differential structure

of these extensions. In particular, we show that the nth iterated

logarithms and their translates are algebraically independent over

the field generated by all lower level iterated logarithms. Our

analysis provides an algorithm for determining the differential

field generated by any rational expression in iterated logarithms.