Extensions by Antiderivatives, Exponentials of Integrals and by Iterated Logarithms

TL;DR

This paper generalizes the Kolchin-Ostrowski theorem for towers of differential field extensions involving J-I-E antiderivatives, including iterated logarithms, and provides an algorithm to classify their finitely generated subfields.

Contribution

It introduces a new proof and generalization of the Kolchin-Ostrowski theorem for J-I-E antiderivative towers and develops an algorithm for subfield classification.

Findings

J-I-E antiderivatives are algebraically independent over the base field.

The generalized theorem applies to towers of extensions by iterated logarithms.

An algorithm for computing finitely differentially generated subfields is presented.

Abstract

Let F be a characteristic zero differential field with an algebraically closed field of constants, E be a no-new-constant extension of F by antiderivatives of F and let y1, ..., yn be antiderivatives of E. The antiderivatives y1, ..., yn of E are called J-I-E antiderivatives if the derivatives of yi in E satisfies certain conditions. We will discuss 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 extensions by iterated logarithms. We will discuss the normality of extensions by iterated logarithms and produce an algorithm to compute its finitely differentially generated subfields.