Complete Reduction for Derivatives in a Transcendental Liouvillian Extension

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 $F$.

💡 Research Summary

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The paper develops a comprehensive theory of complete reduction for derivatives in transcendental Liouvillian extensions, a class of differential fields that can model poly‑logarithmic, hyper‑exponential, trigonometric functions, logarithmic integrals, and nested rational expressions.

Main Contributions

1. Complete Reduction Operator – For a differential field (F_n = C(t_1,\dots ,t_n)) (characteristic zero, constants (C)), the authors construct a linear projector (\varphi_{n,h}) that complements the image of the Risch operator (R_h(y)=y’+hy). The projector satisfies (\varphi_{n,h}^2=\varphi_{n,h}) and (\ker(\varphi_{n,h})=\operatorname{im}(R_h)). Consequently every element (f\in F_n) can be uniquely written as
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