Complete Reduction for Derivatives in a Transcendental Liouvillian Extension
Shaoshi Chen, Hao Du, Yiman Gao, Hui huang, Wenqiao Li, Ziming Li
TL;DR
The paper develops a complete reduction framework for derivatives in transcendental Liouvillian extensions, enabling an additive decomposition $f = g' + r$ with a computable complement that detects elementary integrability. It constructs an inductive, three-step strategy using normalization and companion operators to lift reductions from $F_{n-1}$ to $F_n$, treating both primitive and hyperexponential monomials via auxiliary subspaces and echelon sequences. This yields practical algorithms for in-field and elementary integration, and a reduction-based approach to creative telescoping for functions in such extensions, backed by experimental evidence of improved performance over standard computer algebra systems. The work significantly extends Risch-type reductions to wide classes of Liouvillian towers, with implications for automatic integration and symbolic telescoping in complex symbolic frameworks.
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 $F$.