Complete Reduction for Derivatives in a Transcendental Liouvillian Extension

2026-02-03

AI Summary

This study addresses the effective decidability of whether a given function in a Liouvillian extension is a derivative. By explicitly constructing a complementary subspace to the space of derivatives over the constant field, the authors achieve, for the first time, a unique decomposition of any element into a sum of a derivative part and a remainder term. The proposed complete reduction algorithm precisely determines whether the remainder vanishes—i.e., whether the function is itself a derivative—thereby offering a novel approach to deciding elementary integrability and constructing scaling operators. The method integrates differential algebra, Liouvillian extension theory, and computation of parametric logarithmic parts, significantly advancing structural analysis capabilities in symbolic integration.

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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 $F$.

Problem

• Transcendental Liouvillian extension
• Derivatives
• Elementary integrability
• Telescoper construction
• Differential fields

Innovation

• Transcendental Liouvillian extension
• complementary subspace
• elementary integrability
• telescoper construction
• derivative decomposition