超越Liouvillian扩张中导数的完全约化
Complete Reduction for Derivatives in a Transcendental Liouvillian Extension
AI总结：
针对超越Liouvillian扩张，构造常数子域上导数子空间的补空间，并开发算法将域中任意元素分解为导数与补分量之和，从而判定元素是否为导数，并应用于初等可积性判定与伸缩子的构造。
摘要：
超越Liouvillian扩张是微分域，其中可以建模多对数、超指数和三角函数、对数积分及其（嵌套）有理表达式。对于这样的扩张，我们在常数子域上构造导数子空间的补空间，并开发一种算法，将域中任意元素分解为导数与位于补空间中的分量之和。因此，一个元素是导数当且仅当其补分量为零。此外，该算法使我们能够通过计算参数对数部分来确定扩张上的初等可积性，并在给定先验阶界的情况下，引出一种基于约化的方法来构造扩张中元素的伸缩子。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
