@article{Chen2027JSC,
title = {Complete reduction for derivatives in a transcendental Liouvillian extension},
journal = {Journal of Symbolic Computation},
volume = {140},
pages = {102634},
year = {2027},
issn = {0747-7171},
doi = {https://doi.org/10.1016/j.jsc.2026.102634},
url = {https://www.sciencedirect.com/science/article/pii/S0747717126000829},
author = {Shaoshi Chen and Hao Du and Yiman Gao and Hui Huang and Wenqiao Li and Ziming Li},
keywords = {Complete reduction, Elementary integration, In-field integration, Liouvillian extension, Risch equation, Symbolic integration},
abstract = {Transcendental Liouvillian extensions are differential fields in which one can model polylogarithmic, hyperexponential, and trigonometric functions, logarithmic integrals, and their (nested) rational expressions. For such an extension, we construct a complement, over the subfield of constants, to the subspace of derivatives and develop an algorithm that decomposes any element of the field into the sum of a derivative and a component in this complement. Consequently, an element is a derivative if and only if its complementary component vanishes. Moreover, the algorithm enables us to determine elementary integrability over the extension by computing parametric logarithmic parts. It also yields a reduction-based approach to constructing telescopers for elements of the extension, provided that an a priori order bound is available.}
}