2026/02/03 by Shaoshi Chen, Hao Du, Yiman Gao +3 · 1 voice
Computer Science · #acm:68U01 #cs.SC #msc:68U01
paper · pdf · doi:10.48550/arxiv.2602.03592
49pages
arxiv published 2026/02/03 · arxiv created 2026/08/04 · arxiv updated 2026/08/04
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, we construct, over the subfield of constants, a complement of the subspace of derivatives, and develop an algorithm that decomposes any element of the field into the sum of a derivative and a component lying in the 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, and leads to a reduction-based approach to constructing telescopers for elements in the extension, provided that an a priori order bound is given.