Complete Reduction for Derivatives in a Transcendental Liouvillian Extension

TL;DR

This paper introduces an algorithm for decomposing functions in transcendental Liouvillian extensions into derivatives and a complementary part, facilitating the determination of elementary integrability and the construction of telescopers.

Contribution

It develops a method to explicitly compute a decomposition of functions in transcendental Liouvillian extensions, advancing symbolic integration and telescoping techniques.

Findings

Algorithm computes pair (g, r) such that f = g' + r for functions in F.

Determines elementary integrability by checking if the residual r is zero.

Enables construction of telescopers for functions in the extension.

Abstract

Transcendental Liouvillian extensions are differential fields, in which one can model poly-logarithmic, hyperexponential, and trigonometric functions, logarithmic integrals, and their (nested) rational expressions. For such an extension $(F, \, ^\prime)$ with the subfield $C$ of constants, we construct a complementary subspace $W$ for the $C$-subspace of derivatives in $F$, and develop an algorithm that, for every $f \in F$, computes a pair $(g,r) \in F \times W$ such that $f = g^\prime + r$. Moreover, $f$ is a derivative in $F$ if and only if $r=0$. The algorithm enables us to determine elementary integrability over $F$ by computing parametric logarithmic parts, and leads to a reduction-based approach to constructing telescopers for functions that can be represented by elements in $F$.