| Let F be a characteristic zero differential field with an algebraically closed field of constants C, E ⊃ K ⊃ F be no new constant extensions of F such that K is an extension by antiderivatives of F, and let E contain antiderivatives eta1, ..., eta n of K. The antiderivatives eta1,...,eta n of K are called J-I-E antiderivatives if eta'i ∈ K satisfies 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. |
